{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T12:41:59Z","timestamp":1780404119814,"version":"3.54.1"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T00:00:00Z","timestamp":1669680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basque Government","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we prove the fixed-point theorem for rational contractive mapping on \u00ae-metric space. Additionally, an Euclidean metric space with a binary relation example and an application to the first-order boundary value problem are given. Moreover, the obtained results generalize and extend some of the well-known results in the literature.<\/jats:p>","DOI":"10.3390\/sym14122518","type":"journal-article","created":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T01:42:02Z","timestamp":1669686122000},"page":"2518","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Solving a Boundary Value Problem via Fixed-Point Theorem on \u00ae-Metric Space"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4191-3338","authenticated-orcid":false,"given":"Gunaseelan","family":"Mani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gopinath","family":"Janardhanan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3877-2714","authenticated-orcid":false,"given":"Ozgur","family":"Ege","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ege University, Bornova 35100, Izmir, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arul Joseph","family":"Gnanaprakasam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, Department of Electricity and Electronics, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sure operations dans tes ensembles abstraits et leur application aux equations integrals","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. 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