{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:43:04Z","timestamp":1760229784999,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,27]],"date-time":"2022-06-27T00:00:00Z","timestamp":1656288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of this article is to introduce an ordered implicit relation that can be used for the existence of fixed points of new contractions defined in cone A-metric spaces. We investigate a fixed-point method for proving the existence of Urysohn integral equation solutions. We prove an homotopy result by the application of obtained fixed-point theorem. The hypothesis is demonstrated with examples.<\/jats:p>","DOI":"10.3390\/sym14071328","type":"journal-article","created":{"date-parts":[[2022,6,28]],"date-time":"2022-06-28T00:07:02Z","timestamp":1656374822000},"page":"1328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation"],"prefix":"10.3390","volume":"14","author":[{"given":"Anam","family":"Arif","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1274-1936","authenticated-orcid":false,"given":"Muhammad","family":"Nazam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Allama Iqbal Open University, Islamabad 44000, Pakistan"}]},{"given":"Hamed H.","family":"Al-Sulami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"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":"Hasan","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1090\/S0002-9939-03-07220-4","article-title":"A fixed point theorem in partially ordered sets and some applications to matrix equations","volume":"132","author":"Ran","year":"2004","journal-title":"Proc. Am. Math. 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