{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T06:51:46Z","timestamp":1770965506565,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T00:00:00Z","timestamp":1605830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.<\/jats:p>","DOI":"10.3390\/axioms9040136","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T09:46:18Z","timestamp":1605865578000},"page":"136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Oscillation of Emden\u2013Fowler-Type Neutral Delay Differential Equations"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9740-3081","authenticated-orcid":false,"given":"Shyam Sundar","family":"Santra","sequence":"first","affiliation":[{"name":"Department of Mathematics, JIS College of Engineering, Kalyani 741235, India"}]},{"given":"Taher A.","family":"Nofal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taif University, Taif 21944, KSA, Saudi Arabia"},{"name":"Mathematics Department, Faculty of Science, El- Minia University, El- Minia 61768, Egypt"}]},{"given":"Hammad","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taif University, Taif 21944, KSA, Saudi Arabia"}]},{"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, Seiyun 50512, Yemen"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/0022-247X(78)90104-X","article-title":"Oscillation Theorems for second-order functional-differential equations","volume":"63","author":"Brands","year":"1978","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.camwa.2010.10.035","article-title":"Oscillation Theorems for second-order neutral differential equations","volume":"61","author":"Baculikova","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4472","DOI":"10.1016\/j.camwa.2011.10.024","article-title":"Oscillation Theorems for second-order nonlinear neutral differential equations","volume":"62","author":"Baculikova","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bohner, M., Grace, S.R., and Jadlovska, I. (2017). Oscillation criteria for second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ., 1\u201312.","DOI":"10.14232\/ejqtde.2017.1.60"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.amc.2018.10.091","article-title":"New oscillation criteria for second-order half-linear advanced differential equations","volume":"347","author":"Chatzarakis","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_6","first-page":"170","article-title":"Improved oscillation results for second-order half-linear delay differential equations","volume":"48","author":"Chatzarakis","year":"2019","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_7","first-page":"61","article-title":"Oscillation Theorems for second-order advanced neutral differential equations","volume":"48","year":"2011","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1515\/ms-2017-0003","article-title":"Oscillation of neutral second-order half-linear differential equations without commutativity in delays","volume":"67","author":"Fisnarova","year":"2017","journal-title":"Math. Slovaca"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Umar, M., Sabir, Z., Raja, M.A.Z., Shoaib, M., Gupta, M., and Sanchez, Y.G. (2020). A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. Symmetry, 12.","DOI":"10.3390\/sym12101628"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7359242","DOI":"10.1155\/2020\/7359242","article-title":"Design and numerical solutions of a novel third-order nonlinear Emden\u2013Fowler delay differential model","volume":"2020","author":"Guirao","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2478\/amns.2020.2.00012","article-title":"A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives","volume":"5","author":"Touchent","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"224","DOI":"10.3389\/fphy.2020.00224","article-title":"A Neuro-Swarming Intelligence-Based Computing for Second Order Singular Periodic Non-linear Boundary Value Problems","volume":"8","author":"Sabir","year":"2020","journal-title":"Front. Phys."},{"key":"ref_13","first-page":"1","article-title":"An improved approach for studying oscillation of second-order neutral delay differential equations","volume":"193","author":"Grace","year":"2018","journal-title":"J. Inequ. Appl."},{"key":"ref_14","first-page":"633","article-title":"Oscillation Theorems for second-order nonlinear delay differential equations of neutral type","volume":"48","author":"Karpuz","year":"2019","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s12190-018-1208-6","article-title":"New oscillation criterion for Emden\u2013Fowler type nonlinear neutral delay differential equations","volume":"60","author":"Li","year":"2018","journal-title":"J. Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.37193\/CJM.2014.01.01","article-title":"Oscillation of second-order differential equations with a sublinear neutral term","volume":"30","author":"Agarwal","year":"2014","journal-title":"Carpathian J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2011.1.74","article-title":"Oscillation Theorems for second order neutral differential equations","volume":"74","author":"Baculikova","year":"2011","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., Ruggieri, M., Santra, S.S., and Scapellato, A. (2020). Qualitative Properties of Solutions of Second-Order Neutral Differential Equations. Symmetry, 12.","DOI":"10.3390\/sym12091520"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Li, T., and Rogovchenko, Y.V. (2014). Oscillation Theorems for second-order nonlinear neutral delay differential eqquations. Abstr. Appl. Anal., 594190.","DOI":"10.1155\/2014\/594190"},{"key":"ref_20","first-page":"1","article-title":"Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients","volume":"35","author":"Li","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-018-0506-9","article-title":"Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays","volume":"20","author":"Pinelas","year":"2018","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1515\/anly-2018-0010","article-title":"Necessary and sufficient conditions for oscillation of nonlinear first order forced differential equations with several delays of neutral type","volume":"39","author":"Pinelas","year":"2019","journal-title":"Analysis"},{"key":"ref_23","first-page":"323","article-title":"Some new osciilation criteria for higher order quasi-linear neutral delay differential equations","volume":"3","author":"Qian","year":"2011","journal-title":"Differ. Equ. Appl."},{"key":"ref_24","first-page":"33","article-title":"Existence of positive solution and new oscillation criteria for nonlinear first-order neutral delay differential equations","volume":"8","author":"Santra","year":"2016","journal-title":"Differ. Equ. Appl."},{"key":"ref_25","first-page":"111","article-title":"Oscillation analysis for nonlinear neutral differential equations of second-order with several delays","volume":"59","author":"Santra","year":"2017","journal-title":"Mathematica"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"63","DOI":"10.24193\/mathcluj.2019.1.06","article-title":"Oscillation analysis for nonlinear neutral differential equations of second-order with several delays and forcing term","volume":"61","author":"Santra","year":"2019","journal-title":"Mathematica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"459","DOI":"10.46793\/KgJMat2003.459S","article-title":"Necessary and sufficient condition for oscillatory and asymptotic behavior of second-order functional differential equations","volume":"44","author":"Santra","year":"2020","journal-title":"Krag. J. Math."},{"key":"ref_28","first-page":"1","article-title":"Nonlinear Robin problems with general potential and crossing reaction","volume":"30","author":"Papageorgiou","year":"2019","journal-title":"Rend. Lincei-Mat. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1515\/anona-2020-0009","article-title":"Constant sign and nodal solutions for parametric (p, 2)-equations","volume":"9","author":"Papageorgiou","year":"2020","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"111758","DOI":"10.1016\/j.na.2020.111758","article-title":"Existence of solution to Korteweg-de Vries equation in a non-parabolic domain","volume":"195","author":"Benia","year":"2020","journal-title":"Nonlinear Anal.-Theory Methods Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Benia, Y., Ruggieri, M., and Scapellato, A. (2019). Exact Solutions for a Modified Schr\u00f6dinger Equation. Mathematics, 7.","DOI":"10.3390\/math7100908"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1007\/s10440-014-9924-x","article-title":"Approximate Analysis of a Nonlinear Dissipative Model","volume":"132","author":"Ruggieri","year":"2014","journal-title":"Acta Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Hale, J. (1977). Theory of Functional Differential Equations. Applied Mathematical Sciences, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-9892-2"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/4\/136\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:34:51Z","timestamp":1760178891000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/4\/136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,20]]},"references-count":33,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["axioms9040136"],"URL":"https:\/\/doi.org\/10.3390\/axioms9040136","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,20]]}}}