{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T20:14:02Z","timestamp":1773864842585,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"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>This study aims to investigate the oscillatory behavior of the solutions of an even-order delay differential equation with distributed deviating arguments. We first study the monotonic properties of positive decreasing solutions or the so-called Kneser solutions. Then, by iterative deduction, we improve these properties, which enables us to apply them more than once. Finally, depending on the symmetry between the positive and negative solutions of the studied equation and by combining the new condition for the exclusion of Kneser solutions with some well-known results in the literature, we establish a new standard for the oscillation of the investigated equation.<\/jats:p>","DOI":"10.3390\/sym15020502","type":"journal-article","created":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T02:41:56Z","timestamp":1676342516000},"page":"502","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Higher-Order Delay Differential Equation with Distributed Deviating Arguments: Improving Monotonic Properties of Kneser Solutions"],"prefix":"10.3390","volume":"15","author":[{"given":"Shaimaa","family":"Elsaeed","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, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia"},{"name":"Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia"}]},{"given":"Elmetwally M.","family":"Elabbasy","sequence":"additional","affiliation":[{"name":"Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Braun, M., and Golubitsky, M. (1983). Differential Equations and Their Applications, Springer.","DOI":"10.1007\/978-1-4684-0164-6"},{"key":"ref_2","unstructured":"Heinmets, F. (1969). Concept and Models of Biomathematics, Marcel Dekker."},{"key":"ref_3","unstructured":"Zachmanoglou, E.C., and Thoe, D.W. (1986). Introduction to Partial Differential Equations with Applications, Courier Corporation."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"125192","DOI":"10.1016\/j.amc.2020.125192","article-title":"New oscillation criteria for nonlinear delay differential equations of fourth-order","volume":"377","author":"Moaaz","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3272","DOI":"10.3934\/math.2021196","article-title":"Neutral differential equations with noncanonical operator: Oscillation behavior of solutions","volume":"6","author":"Elabbasy","year":"2021","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Moaaz, O., Baleanu, D., and Muhib, A. (2020). New aspects for non-existence of kneser solutions of neutral differential equations with odd-order. 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Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.amc.2008.09.021","article-title":"Oscillation criteria for second-order neutral equations with distributed deviating argument","volume":"206","author":"Zhao","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-019-2060-1","article-title":"On the oscillation of fourth-order delay differential equations","volume":"2019","author":"Grace","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Moaaz, O., and Cesarano, C. (2021). New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application. Mathematics, 9.","DOI":"10.3390\/math9161971"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Muhib, A., Abdeljawad, T., Moaaz, O., and Elabbasy, E.M. (2020). Oscillatory properties of odd-order delay differential equations with distribution deviating arguments. Appl. Sci., 10.","DOI":"10.3390\/app10175952"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.camwa.2009.06.027","article-title":"Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients","volume":"59","author":"Zhang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1186\/s13662-017-1312-1","article-title":"On the asymptotic behavior of fourth-order functional differential equations","volume":"2017","author":"Moaaz","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","first-page":"235","article-title":"Some new oscillation criteria for fourth-order neutral differential equations with distributed delay","volume":"7","author":"Tunc","year":"2019","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_18","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 delays","volume":"36","author":"Philos","year":"1981","journal-title":"Arch. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2000). Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic.","DOI":"10.1007\/978-94-015-9401-1"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10958-012-1071-1","article-title":"On the oscillation of higher-order delay differential equations","volume":"187","author":"Graef","year":"2012","journal-title":"J. Math. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1090\/S0002-9939-1980-0548086-5","article-title":"Oscillation of first-order nonlinear differential equations with deviating arguments","volume":"78","author":"Kitamura","year":"1980","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"227","DOI":"10.7494\/OpMath.2020.40.2.227","article-title":"On the asymptotic behavior of non-oscillatory solutions of certain fractional differential equations with positive and negative terms","volume":"40","author":"Graef","year":"2020","journal-title":"Opusc. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1515\/gmj-2017-0026","article-title":"On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations","volume":"25","author":"Grace","year":"2018","journal-title":"Georgian Math. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1007\/s00009-018-1120-1","article-title":"On the asymptotic behavior of non-oscillatory solutions of certain fractional differential equations","volume":"15","author":"Grace","year":"2018","journal-title":"Mediterr. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Alzabut, J., Agarwal, R.P., Grace, S.R., and Jonnalagadda, J.M. (2022). Oscillation results for solutions of fractional-order differential equations. 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