{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:17:39Z","timestamp":1760149059481,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,2]],"date-time":"2023-07-02T00:00:00Z","timestamp":1688256000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results\u2019 significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays.<\/jats:p>","DOI":"10.3390\/axioms12070658","type":"journal-article","created":{"date-parts":[[2023,7,3]],"date-time":"2023-07-03T00:53:16Z","timestamp":1688345596000},"page":"658","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4643-8318","authenticated-orcid":false,"given":"Fahd","family":"Masood","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, Corso Vittorio Emanuele II, 39, 00186 Roma, 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","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"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Braun, M. (1993). Qualitative Theory of Differential Equations: Differential Equations and Their Applications, Springer.","DOI":"10.1007\/978-1-4612-4360-1_4"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cooke, K.L. (1963). 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