{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:26:12Z","timestamp":1760145972311,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T00:00:00Z","timestamp":1727049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","doi-asserted-by":"publisher","award":["PNURSP2024R406"],"award-info":[{"award-number":["PNURSP2024R406"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","award":["PNURSP2024R406"],"award-info":[{"award-number":["PNURSP2024R406"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method\u2019s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results.<\/jats:p>","DOI":"10.3390\/axioms13090652","type":"journal-article","created":{"date-parts":[[2024,9,24]],"date-time":"2024-09-24T08:56:06Z","timestamp":1727168166000},"page":"652","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-1022","authenticated-orcid":false,"given":"Osama","family":"Moaaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia"}]},{"given":"Shaimaa","family":"Elsaeed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"given":"Asma","family":"Al-Jaser","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Samia","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Department of Basic Science, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt"}]},{"given":"Amira","family":"Essam","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1971). Functional Differential Equations, in Analytic Theory of Differential Equations, Springer.","DOI":"10.1007\/978-1-4615-9968-5"},{"key":"ref_2","unstructured":"MacDonald, N. (1989). Biological Delay Systems: Linear Stability Theory, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rihan, F.A. (2021). Delay Differential Equations and Applications to Biology, Springer.","DOI":"10.1007\/978-981-16-0626-7"},{"key":"ref_4","first-page":"885","article-title":"Chen-Charpentier, Random coefficient differential equation models for bacterial growth","volume":"50","author":"Stanescu","year":"2009","journal-title":"Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"e22684","DOI":"10.1002\/num.22684","article-title":"A numerical study on fractional differential equation with population growth model","volume":"40","author":"Kumar","year":"2020","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17","DOI":"10.2307\/1939378","article-title":"Partial differential equations in ecology: Spatial interactions and population dynamics","volume":"75","author":"Holmes","year":"1994","journal-title":"Ecology"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.apnum.2022.04.003","article-title":"Effective condition number for the selection of the RBF shape parameter with the fictitious point method","volume":"178","author":"Noorizadegan","year":"2022","journal-title":"Appl. Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"105933","DOI":"10.1016\/j.enganabound.2024.105933","article-title":"Stable weight updating: A key to reliable PDE solutions using deep learning","volume":"168","author":"Noorizadegan","year":"2024","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1090\/S0002-9947-1918-1501107-2","article-title":"Concerning the zeros of the solutions of certain differential equations","volume":"19","author":"Fite","year":"1918","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2002). Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-2515-6"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2003). Oscillation Theory for Second Order Dynamic Equations, Taylor & Francis, Ltd.","DOI":"10.4324\/9780203222898"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Bohner, M., and Li, W.-T. (2004). Nonoscillation and oscillation: Theory for functional differential equations. Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc.","DOI":"10.1201\/9780203025741"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gyori, I., and Ladas, G. (1991). Oscillation Theory of Delay Differential Equations, Oxford University Press. Oxford Mathematical Monographs.","DOI":"10.1093\/oso\/9780198535829.001.0001"},{"key":"ref_14","unstructured":"Erbe, L.H., Kong, Q., and Zhang, B.G. (1995). Oscillation Theory for Functional Differential Equations, Dekker."},{"key":"ref_15","unstructured":"Shames, I.H., and Pitarresi, J.M. (1989). Introduction to Solid Mechanics, Springer."},{"key":"ref_16","first-page":"1","article-title":"An Overview of Mathematical Methods Applied in the Biomechanics of Foot and Ankle\u2013Foot Orthosis Models","volume":"7","author":"Nazha","year":"2023","journal-title":"J"},{"key":"ref_17","unstructured":"Fung, Y.C. (2013). Biomechanics. Mechanical Properties of Living Tissues, Springer Science & Business Media."},{"key":"ref_18","unstructured":"Smith, W.F. (2010). Waves and Oscillations: A Prelude to Quantum Mechanics, Oxford University Press."},{"key":"ref_19","unstructured":"Fowler, A.C. (1997). Mathematical Models in the Applied Sciences, Cambridge University Press."},{"key":"ref_20","unstructured":"Strogatz, S.H. (1996). Nonlinear Dynamics and Chaoss: With Applications to Physics, Biology, Chemistry, and Engineering, CRC Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/BF01444165","article-title":"Untersuchungen \u00fcber die reellen Nullstellen der Integrale linearer Differentialgleichungen","volume":"42","author":"Kneser","year":"1893","journal-title":"Math. Ann."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"627","DOI":"10.2307\/2372355","article-title":"Oscillatory and non-oscillatory linear differential equations","volume":"71","author":"Hartman","year":"1949","journal-title":"Am. J. Math."},{"key":"ref_23","unstructured":"Agarwal, R.P., Grace, S.R., and O\u2019Regan, D. (2013). Oscillation Theory for Difference and Functional Differential Equations, Springer Science & Business Media."},{"key":"ref_24","first-page":"510","article-title":"Oscillations of first-order neutral delay differential equations","volume":"120","author":"Grammatikopoulos","year":"1986","journal-title":"J. Math. Appl."},{"key":"ref_25","unstructured":"Ladde, G.S., Lakshmikantham, V., and Zhang, B.G. (1987). Oscillation Theory of Differential Equations with Deviating Argument, Marcel Dekker."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"368","DOI":"10.2307\/2372182","article-title":"On the non-existence of conjugate points","volume":"73","author":"Wintner","year":"1951","journal-title":"Am. J. Math."},{"key":"ref_27","first-page":"2465","article-title":"Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete Contin","volume":"13","author":"Moaaz","year":"2020","journal-title":"Dyn. Syst. Ser. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/BF02192675","article-title":"On second order sublinear oscillation","volume":"27","author":"Philos","year":"1984","journal-title":"Aequationes Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF02415027","article-title":"Nonlinear oscillation of second order differential equations with retarded argument","volume":"106","author":"Kusano","year":"1975","journal-title":"Ann. Mat. Pura Ed Applicata"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"28051","DOI":"10.3934\/math.20231435","article-title":"Improved results for testing the oscillation of functional differential equations with multiple delays","volume":"8","author":"Essam","year":"2023","journal-title":"AIMS Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8853745","DOI":"10.1155\/2020\/8853745","article-title":"Second-order differential equation with multiple delays: Oscillation theorems and applications","volume":"2020","author":"Santra","year":"2020","journal-title":"Complexity"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1007\/s40840-014-0048-2","article-title":"Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators","volume":"38","author":"Zhang","year":"2015","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Santra, S.S., Sethi, A.K., Moaaz, O., Khedher, K.M., and Yao, S.W. (2021). New oscillation theorems for second-order differential equations with canonical and non-canonical operator via riccati transformation. Mathematics, 9.","DOI":"10.3390\/math9101111"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"31","DOI":"10.21136\/MB.2009.140637","article-title":"Oscillation of second order neutral delay differential equations","volume":"134","author":"Hudakova","year":"2009","journal-title":"Math. Bohem."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1186\/1687-1847-2012-227","article-title":"New oscillation results for second-order neutral delay dynamic equations","volume":"2012","author":"Zhang","year":"2012","journal-title":"Adv. Differ. Equations"},{"key":"ref_36","first-page":"125289","article-title":"Kneser-type oscillation criteria for second-order half-linear delay differential equations","volume":"380","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1007\/s10958-014-1990-0","article-title":"Oscillation of fourth-order delay differential equations","volume":"201","author":"Zhang","year":"2014","journal-title":"J. Math. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s12190-008-0158-9","article-title":"Oscillation theorems for fourth order functional differential equations","volume":"30","author":"Grace","year":"2009","journal-title":"J. Appl. Math. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1186\/s13662-022-03699-4","article-title":"Half-linear differential equations of fourth order: Oscillation criteria of solutions","volume":"2022","author":"Bazighifan","year":"2022","journal-title":"Adv. Contin. Discret. Model."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"118","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_41","doi-asserted-by":"crossref","unstructured":"Chatzarakis, G., Elabbasy, E., and Bazighifan, O. (2019). An oscillationcriterion in 4th-order neutral differential equations with a con tinuously distributed delay. Adv. Differ. Equ., 336.","DOI":"10.1186\/s13662-019-2281-3"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"106940","DOI":"10.1016\/j.aml.2020.106940","article-title":"Oscillation of nth order strongly noncanonical delay differential equations","volume":"115","author":"Dzurina","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s11253-007-0021-4","article-title":"Oscillation of certain fourth order functional differential equations","volume":"59","author":"Agarwal","year":"2007","journal-title":"Ukr. Math. J."},{"key":"ref_44","first-page":"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_45","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF02413181","article-title":"Nonlinear oscillation of fourth order functional differential equations","volume":"119","author":"Onose","year":"1979","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Almarri, B., Janaki, S., Ganesan, V., Ali, A.H., Nonlaopon, K., and Bazighifan, O. (2022). Novel oscillation theorems and symmetric properties of nonlinear delay differential equations of fourth-order with a middle term. Symmetry, 14.","DOI":"10.3390\/sym14030585"},{"key":"ref_47","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_48","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_49","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1017\/S0017089509990188","article-title":"Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations","volume":"52","author":"Karpuz","year":"2010","journal-title":"Glasgow Math. J."},{"key":"ref_50","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. Comput."},{"key":"ref_51","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."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1007\/BF01223686","article-title":"Ot 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_53","doi-asserted-by":"crossref","first-page":"509","DOI":"10.2969\/jmsj\/03330509","article-title":"Comparison theorems for functional-differential equations with deviating arguments","volume":"33","author":"Kusano","year":"1981","journal-title":"J. Math. Soc. Jpn."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/652\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:02:29Z","timestamp":1760112149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/652"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,23]]},"references-count":53,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090652"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090652","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,9,23]]}}}