{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T04:10:01Z","timestamp":1771647001797,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Research and Development Project of Hainan Province","award":["ZDYF2023GXJS007"],"award-info":[{"award-number":["ZDYF2023GXJS007"]}]},{"name":"Key Research and Development Project of Hainan Province","award":["621RC602"],"award-info":[{"award-number":["621RC602"]}]},{"name":"Key Research and Development Project of Hainan Province","award":["2022KJCX22"],"award-info":[{"award-number":["2022KJCX22"]}]},{"name":"Key Research and Development Project of Hainan Province","award":["USY22XK-04"],"award-info":[{"award-number":["USY22XK-04"]}]},{"name":"High-Level Project of Hainan Provincial Natural Science Foundation","award":["ZDYF2023GXJS007"],"award-info":[{"award-number":["ZDYF2023GXJS007"]}]},{"name":"High-Level Project of Hainan Provincial Natural Science Foundation","award":["621RC602"],"award-info":[{"award-number":["621RC602"]}]},{"name":"High-Level Project of Hainan Provincial Natural Science Foundation","award":["2022KJCX22"],"award-info":[{"award-number":["2022KJCX22"]}]},{"name":"High-Level Project of Hainan Provincial Natural Science Foundation","award":["USY22XK-04"],"award-info":[{"award-number":["USY22XK-04"]}]},{"name":"Sanya City Science and Technology Innovation Special Project","award":["ZDYF2023GXJS007"],"award-info":[{"award-number":["ZDYF2023GXJS007"]}]},{"name":"Sanya City Science and Technology Innovation Special Project","award":["621RC602"],"award-info":[{"award-number":["621RC602"]}]},{"name":"Sanya City Science and Technology Innovation Special Project","award":["2022KJCX22"],"award-info":[{"award-number":["2022KJCX22"]}]},{"name":"Sanya City Science and Technology Innovation Special Project","award":["USY22XK-04"],"award-info":[{"award-number":["USY22XK-04"]}]},{"name":"Key Special Project of University of Sanya","award":["ZDYF2023GXJS007"],"award-info":[{"award-number":["ZDYF2023GXJS007"]}]},{"name":"Key Special Project of University of Sanya","award":["621RC602"],"award-info":[{"award-number":["621RC602"]}]},{"name":"Key Special Project of University of Sanya","award":["2022KJCX22"],"award-info":[{"award-number":["2022KJCX22"]}]},{"name":"Key Special Project of University of Sanya","award":["USY22XK-04"],"award-info":[{"award-number":["USY22XK-04"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, our main aim is to present innovative fixed-point theorems that provide solutions to the fixed-circle problem with symmetric contractions. We accomplish this by employing operator enrichment techniques within the context of Banach spaces. Furthermore, we demonstrate the practical application of these theorems by showcasing their relevance to the rectified linear unit (ReLU) activation function. By exploring the connection between fixed points and activation functions, our work contributes to a deeper understanding of the behavior and properties of these fundamental mathematical concepts.<\/jats:p>","DOI":"10.3390\/sym16010069","type":"journal-article","created":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T09:47:32Z","timestamp":1704361652000},"page":"69","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Application to Activation Functions through Fixed-Circle Problems with Symmetric Contractions"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2864-6730","authenticated-orcid":false,"given":"Rizwan","family":"Anjum","sequence":"first","affiliation":[{"name":"Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan"}]},{"given":"Mujahid","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan"}]},{"given":"Hira","family":"Safdar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2396-2943","authenticated-orcid":false,"given":"Muhammad","family":"Din","sequence":"additional","affiliation":[{"name":"Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6496-4824","authenticated-orcid":false,"given":"Mi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Center for Mathematical Research, University of Sanya, Sanya 572022, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8254-6688","authenticated-orcid":false,"given":"Stojan","family":"Radenovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1007\/s40840-017-0555-z","article-title":"Some fixed-circle theorems on metric spaces","volume":"42","year":"2019","journal-title":"Bull. 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