{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T12:39:00Z","timestamp":1780403940898,"version":"3.54.1"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basque Government","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce the concept of fuzzy-controlled bipolar metric space and prove some fixed-point theorems in this space. Our results generalize and expand some of the literature\u2019s well-known results. We also provide some applications of our main results to integral equations.<\/jats:p>","DOI":"10.3390\/axioms12040396","type":"journal-article","created":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T02:01:11Z","timestamp":1681956071000},"page":"396","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fixed-Point Theorems for Nonlinear Contraction in Fuzzy-Controlled Bipolar Metric Spaces"],"prefix":"10.3390","volume":"12","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6346-605X","authenticated-orcid":false,"given":"Arul Joseph","family":"Gnanaprakasam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur, Kanchipuram 603203, Tamil Nadu, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Santosh","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Natural and Applied Sciences, University of Dar es Salaam, Dar es Salaam 35063, Tanzania"}],"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, Izmir 35100, Turkey"}],"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":[[2023,4,19]]},"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. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_3","first-page":"326","article-title":"Fuzzy metric and statistical metric spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/0165-0114(88)90064-4","article-title":"Fixed points in fuzzy metric spaces","volume":"27","author":"Grabiec","year":"1988","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0165-0114(00)00088-9","article-title":"On fixed point theorems in fuzzy metric spaces","volume":"125","author":"Gregori","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0165-0114(94)90162-7","article-title":"On some results in fuzzy metric spaces","volume":"64","author":"George","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5362","DOI":"10.22436\/jnsa.009.09.05","article-title":"Bipolar metric spaces and some fixed point theorems","volume":"9","author":"Mutlu","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"196","DOI":"10.22436\/jnsa.013.04.04","article-title":"Some fixed point theorems in fuzzy bipolar metric spaces","volume":"13","author":"Bartwal","year":"2020","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"371","DOI":"10.24193\/fpt-ro.2022.1.23","article-title":"Generalized quasi-contractions on weak orthogonal metric spaces","volume":"23","author":"Senapati","year":"2022","journal-title":"FPT"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1002\/num.22541","article-title":"Controlled fuzzy metric spaces and some related fixed point results","volume":"37","author":"Sezen","year":"2021","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"224","DOI":"10.26637\/mjm1003\/005","article-title":"A new fixed point result in bipolar controlled fuzzy metric spaces with application","volume":"10","author":"Tiwari","year":"2022","journal-title":"Malaya J. Mat. (MJM)"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.15388\/NA.2016.2.5","article-title":"Multivalued generalizations of fixed point results in fuzzy metric spaces","volume":"21","year":"2016","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_13","first-page":"9129992","article-title":"Solving an Integral Equation by Using Fixed Point Approach in Fuzzy Bipolar Metric Spaces","volume":"2021","author":"Gunaseelan","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_14","first-page":"5044181","article-title":"On orthogonal coupled fixed point results with application","volume":"2022","author":"Mani","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mani, G., Gnanaprakasam, A.J., Mitrovi\u0107, Z.D., and Bota, M.F. (2021). Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces. Mathematics, 9.","DOI":"10.3390\/math9243181"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1016\/j.fss.2007.07.006","article-title":"Fuzzy \u03c8-contractive mappings in non-Archimedean fuzzy metric spaces","volume":"159","author":"Mihet","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1186\/s13660-020-02371-3","article-title":"Some new fixed point results in b-fuzzy metric spaces","volume":"2020","author":"Mukheimer","year":"2020","journal-title":"Inequal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Raki\u0107, D., Do\u0161enovi\u0107, T., Mitrovi\u0107, Z.D., de la Sen, M., and Radenovi\u0107, S. (2020). Some new fixed point results on \u0106iri\u0107 type in fuzzy metric spaces. Mathematics, 8.","DOI":"10.3390\/math8020297"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1515\/ms-2017-0115","article-title":"Suzuki-type of common fixed point theorems in fuzzy metric spaces","volume":"68","author":"Sedghi","year":"2018","journal-title":"Math. Slovaca"},{"key":"ref_20","first-page":"4429173","article-title":"Unique fixed-point results in fuzzy metric spaces with an application to fredholm integral equations","volume":"2021","author":"Shamas","year":"2021","journal-title":"J. Funct. Spaces"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/396\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:19:08Z","timestamp":1760123948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/396"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,19]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040396"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040396","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,19]]}}}