{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:33:43Z","timestamp":1760150023620,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,30]],"date-time":"2023-09-30T00:00:00Z","timestamp":1696032000000},"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 prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results.<\/jats:p>","DOI":"10.3390\/axioms12100941","type":"journal-article","created":{"date-parts":[[2023,10,2]],"date-time":"2023-10-02T04:39:30Z","timestamp":1696221570000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Common Fixed Point Results on a Double-Controlled Metric Space for Generalized Rational-Type Contractions with Application"],"prefix":"10.3390","volume":"12","author":[{"given":"Khaleel","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Ghulam","family":"Murtaza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Salha","family":"Alshaikey","sequence":"additional","affiliation":[{"name":"Saudi Arabia-Alqunfudah Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca 21421, Saudi Arabia"}]},{"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"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,30]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, M.H., and Trujillo, J.J. 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