{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T12:42:16Z","timestamp":1771677736988,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,20]],"date-time":"2022-10-20T00:00:00Z","timestamp":1666224000000},"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 manuscript, we defined (\u03b1, F)-contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are generalizations of a few well-known findings in the literature. Some non-trivial examples and certain consequences are also provided to illustrate the significance of the presented results. The existence and uniqueness of the solution of non-linear fractional differential equations and the monotone iterative method are also determined using the fixed-point method.<\/jats:p>","DOI":"10.3390\/axioms11100573","type":"journal-article","created":{"date-parts":[[2022,10,20]],"date-time":"2022-10-20T20:35:55Z","timestamp":1666298155000},"page":"573","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Reich-Type and (\u03b1, F)-Contractions in Partially Ordered Double-Controlled Metric-Type Spaces with Applications to Non-Linear Fractional Differential Equations and Monotonic Iterative Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Muhammad","family":"Farhan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, H10, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5228-1073","authenticated-orcid":false,"given":"Umar","family":"Ishtiaq","sequence":"additional","affiliation":[{"name":"Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7284-6908","authenticated-orcid":false,"given":"Muhammad","family":"Saeed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7742-5993","authenticated-orcid":false,"given":"Aftab","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"given":"Hamed","family":"Al Sulami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,20]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta Math. Inform. Univ. Ostrav."},{"key":"ref_2","first-page":"47","article-title":"\u03b1-Contractive mappings on rectangular b-metric spaces and an application to integral equations","volume":"9","author":"Alharbi","year":"2018","journal-title":"J. Math. 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