{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T00:14:34Z","timestamp":1779927274849,"version":"3.53.1"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T00:00:00Z","timestamp":1672876800000},"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 paper is to present some new contraction mappings via control functions. In addition, some fixed point results for \u0398,\u03b1,\u03b8,\u03a8 contraction, rational \u0398,\u03b1,\u03b8,\u03a8 contraction and almost \u0398,\u03b1,\u03b8,\u03a8 contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of \u0398,\u03b8,\u03a8\u03c50, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated.<\/jats:p>","DOI":"10.3390\/sym15010164","type":"journal-article","created":{"date-parts":[[2023,1,6]],"date-time":"2023-01-06T01:48:19Z","timestamp":1672969699000},"page":"164","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Control Functions in G-Metric Spaces: Novel Methods for \u03b8-Fixed Points and \u03b8-Fixed Circles with an Application"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-9367","authenticated-orcid":false,"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maryam G.","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ayman","family":"Shehata","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"key":"ref_1","first-page":"289","article-title":"A new approach to generalized metric spaces","volume":"7","author":"Mustafa","year":"2006","journal-title":"J. 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