{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T22:33:01Z","timestamp":1774737181213,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"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>In this paper, we modify various contractive conditions (C.C.)s such as Ciric type (C.C.), Rhoades type (C.C.), Seghal type (C.C.), Bianchini type (C.C.), and Berinde type (C.C.) for two self-mappings, considering that the contractive property plays a major role in establishing a fixed circle (F.C.) on both metric spaces (M-s) and S-(M-s) where the symmetry condition is satisfied, and we utilize them to establish a common (F.C.). We prove new (F.C.) results on both (M-s) and S-(M-s) with illustrative examples. Finally, we provide an application to activation functions such as rectified linear unit activation functions and parametric rectified linear unit activation functions.<\/jats:p>","DOI":"10.3390\/sym15050971","type":"journal-article","created":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T06:20:42Z","timestamp":1682317242000},"page":"971","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Some Common Fixed Circle Results on Metric and \ud835\udd4a-Metric Spaces with an Application to Activation Functions"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4535-4019","authenticated-orcid":false,"given":"Nihal","family":"Ta\u015f","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bal\u0131kesir University, 10145 Bal\u0131kesir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7620-3387","authenticated-orcid":false,"given":"Elif","family":"Kaplan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ondokuz May\u0131s University, 55200 Samsun, T\u00fcrkiye"}]},{"given":"Dania","family":"Santina","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7492-4933","authenticated-orcid":false,"given":"Wasfi","family":"Shatanawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa 13133, Jordan"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"ref_1","first-page":"267","article-title":"A generalization of Banach\u2019s contraction principle","volume":"45","author":"Ciric","year":"1974","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","first-page":"19","article-title":"Generalized contractions and fixed-point theorems","volume":"12","author":"Ciric","year":"1971","journal-title":"Publ. L\u2019Institut Matheematique"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/S0002-9947-1977-0433430-4","article-title":"A comparison of various definitions of contractive mappings","volume":"226","author":"Rhoades","year":"1977","journal-title":"Trans. Am. Math. 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