{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:39:19Z","timestamp":1759970359774,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T00:00:00Z","timestamp":1736985600000},"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 present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We provide different concrete conditions on the real-valued functions \u03a9,\u03a0:0,\u221e\u2192R for the existence of fixed points via the (\u03a9,\u03a0)-contraction in bi-polar metric spaces. Further, we define real-valued functions \u03a9,\u03a0:(0,1]\u2192R to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply \u03a9,\u03a0 fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results.<\/jats:p>","DOI":"10.3390\/sym17010127","type":"journal-article","created":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T10:44:23Z","timestamp":1737024263000},"page":"127","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Existence and Uniqueness of a Solution of a Boundary Value Problem Used in Chemical Sciences via a Fixed Point Approach"],"prefix":"10.3390","volume":"17","author":[{"given":"Umar","family":"Ishtiaq","sequence":"first","affiliation":[{"name":"Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Fahad","family":"Jahangeer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}]},{"given":"Mubariz","family":"Garayev","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh P.O. Box 2455, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8042-1806","authenticated-orcid":false,"given":"Ioan-Lucian","family":"Popa","sequence":"additional","affiliation":[{"name":"Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"},{"name":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.2140\/pjm.1960.10.313","article-title":"Statistical metric spaces","volume":"10","author":"Schweizer","year":"1960","journal-title":"Pac. J. 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