{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T03:26:46Z","timestamp":1774668406223,"version":"3.50.1"},"reference-count":13,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T00:00:00Z","timestamp":1556582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taibah University, Al Madina Al Munawara, Kingdom of Saudi Arabia","award":["60348\/1439"],"award-info":[{"award-number":["60348\/1439"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this study is to investigate the existence of solutions for a non-linear neutral differential equation with an unbounded delay. To achieve our goals, we take advantage of fixed point theorems for self-mappings satisfying a generalized (    \u03b1 , \u03c6    ) rational contraction, as well as cyclic contractions in the context of    F   -metric spaces. We also supply an example to support the new theorem.<\/jats:p>","DOI":"10.3390\/sym11050602","type":"journal-article","created":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T03:15:22Z","timestamp":1556766922000},"page":"602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["New Fixed Point Theorems with Applications to Non-Linear Neutral Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Laila A.","family":"Alnaser","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Taibah University, Al Madina Al Munawara 41411, Saudi Arabia"}]},{"given":"Jamshaid","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1984-9472","authenticated-orcid":false,"given":"Durdana","family":"Lateef","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Taibah University, Al Madina Al Munawara 41411, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6299-9572","authenticated-orcid":false,"given":"Hoda A.","family":"Fouad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Taibah University, Al Madina Al Munawara 41411, Saudi Arabia"},{"name":"Faculty of Science, Alexandria University, Alexandria 21500, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03018603","article-title":"Sur quelques points du calcul fonctionnel","volume":"22","author":"Frechet","year":"1906","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"ref_2","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta Math. 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Methods Nonlinear Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1515\/GMJ.2006.25","article-title":"Fixed point techniques and stability for natural nonlinear differential equations with unbounded delays","volume":"13","author":"Djoudi","year":"2006","journal-title":"Georgian Math. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.trmi.2018.08.006","article-title":"Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results","volume":"172","author":"Hussain","year":"2018","journal-title":"Trans. A Razmadze Math. 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