{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T23:43:19Z","timestamp":1778542999912,"version":"3.51.4"},"reference-count":40,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:00:00Z","timestamp":1701302400000},"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>This article introduces the novel concept of an extended parametric Sb-metric space, which is a generalization of both Sb-metric spaces and parametric Sb-metric spaces. Within this extended framework, we first establish an analog version of the Banach fixed-point theorem for self-maps. We then prove an improved version of the Banach contraction principle for symmetric extended parametric Sb-metric spaces, using an auxiliary function to establish the desired result. Finally, we provide illustrative examples and an application for determining solutions to Fredholm integral equations, demonstrating the practical implications of our work.<\/jats:p>","DOI":"10.3390\/sym15122136","type":"journal-article","created":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T03:58:02Z","timestamp":1701403082000},"page":"2136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fixed Point Theory in Extended Parametric Sb-Metric Spaces and Its Applications"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7131-2664","authenticated-orcid":false,"given":"Naveen","family":"Mani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Mohali 140413, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8802-2391","authenticated-orcid":false,"given":"Sunil","family":"Beniwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Mohali 140413, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9835-0935","authenticated-orcid":false,"given":"Rahul","family":"Shukla","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9387-6380","authenticated-orcid":false,"given":"Megha","family":"Pingale","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sandip University, Nashik 422213, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03018603","article-title":"Sur quelques points du calcul fonctionnel","volume":"22","year":"1906","journal-title":"Rend. 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