{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:53:14Z","timestamp":1760233994085,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T00:00:00Z","timestamp":1615334400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Function of China","award":["12061062"],"award-info":[{"award-number":["12061062"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with the existence of \u03c9-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t\u2212\u03c4)),t\u2208R, where Lnu(t):=u(n)(t)+\u2211i=0n\u22121aiu(i)(t), ai\u2208R, i=0,1,\u22ef,n\u22121, are constants, f(t,x,y):R\u00d7E\u00d7E\u27f6E is continuous and \u03c9-periodic with respect to t, \u03c4&gt;0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.<\/jats:p>","DOI":"10.3390\/sym13030449","type":"journal-article","created":{"date-parts":[[2021,3,10]],"date-time":"2021-03-10T20:51:42Z","timestamp":1615409502000},"page":"449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9273-1473","authenticated-orcid":false,"given":"He","family":"Yang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5193-030X","authenticated-orcid":false,"given":"Yongxiang","family":"Li","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1006\/jmaa.1994.1250","article-title":"The method of lower and upper solutions for second, third, forth, and higher order boundary value problems","volume":"185","author":"Cabada","year":"1994","journal-title":"J. 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Nonlinear Problems in Abstract Cones, Academic Press."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1016\/0362-546X(83)90006-8","article-title":"On the behaviour of measure of noncompactness with respect to differential and integration of vector-valued functions","volume":"7","author":"Heinz","year":"1983","journal-title":"Nonlinear Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/449\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:35:55Z","timestamp":1760160955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/449"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,10]]},"references-count":14,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030449"],"URL":"https:\/\/doi.org\/10.3390\/sym13030449","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,3,10]]}}}