{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:40:46Z","timestamp":1760240446834,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T00:00:00Z","timestamp":1560902400000},"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 the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods.<\/jats:p>","DOI":"10.3390\/sym11060812","type":"journal-article","created":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T10:43:32Z","timestamp":1560941012000},"page":"812","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions"],"prefix":"10.3390","volume":"11","author":[{"given":"Parbati","family":"Saha","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shantau","family":"Guria","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Binayak S.","family":"Choudhury","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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 IIDP, Faculty of Science and Technology, University of the Basque Country, PO Box 644, de Bilbao, Barrio Sarriena, 48940 Leioa, Bizkaia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topics in Metric Fixed Point Theory (Cambridge Studies in Advanced Mathematics), Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Khamsi, M.A., and Kirk, W.A. (2001). An Introduction to Metric Spaces and Fixed Point Theory, John Wiley Sons.","DOI":"10.1002\/9781118033074"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kirk, W.A., and Sims, B. (2001). Handbook of Metric Fixed Point Theory, Springer.","DOI":"10.1007\/978-94-017-1748-9"},{"key":"ref_4","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_5","first-page":"326","article-title":"Fuzzy metric and statistical metric spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0165-0114(94)90162-7","article-title":"On some result in fuzzy metric space","volume":"64","author":"George","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.fss.2012.07.012","article-title":"Coupled coincidence point results for compatible mappings in partially ordered fuzzy metric spaces","volume":"222","author":"Choudhury","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","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_9","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_10","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1016\/j.fss.2006.11.012","article-title":"On fuzzy contractive mappings in fuzzy metric spaces","volume":"158","author":"Mihet","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","first-page":"287","article-title":"A new contractive mapping principle in fuzzy metric spaces","volume":"8","author":"Saha","year":"2016","journal-title":"Bull. dell\u2019Uni. Math. Ital."},{"key":"ref_12","unstructured":"Kaufman, A., and Gupta, M.M. (1988). Fuzzy Mathematical Models in Engineering Management Science, Elsevier Science Inc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3790","DOI":"10.1016\/j.na.2007.10.014","article-title":"Best proximity points for cyclic Mier-Keeler contractions","volume":"69","author":"Bari","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_14","first-page":"41","article-title":"A best proximity point theorem using discontinuous functions","volume":"24","author":"Choudhury","year":"2017","journal-title":"J. Convex Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1016\/j.bulsci.2013.02.003","article-title":"Best proximity points for \u03b1-\u03c8-proximal contractive type mappings and applications","volume":"137","author":"Jleli","year":"2013","journal-title":"Bull. Sci. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"193085","DOI":"10.1155\/2012\/193085","article-title":"Best proximity point result for MK-proximal contractions","volume":"2012","author":"Jleli","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1016\/j.aml.2012.02.008","article-title":"Best proximity points of cyclic mappings","volume":"25","author":"Karapinar","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_18","first-page":"447","article-title":"Best proximity point theorems for non-self mappings","volume":"14","author":"Raj","year":"2013","journal-title":"Fixed Point Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4804","DOI":"10.1016\/j.na.2011.04.052","article-title":"A best proximity point theorem for weakly contractive non-self-mappings","volume":"74","author":"Raj","year":"2011","journal-title":"Nonlinear Anal."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"De la Sen, M., Abbas, M., and Saleem, N. (2017). On Optimal Fuzzy best proximity coincidence points of proximal contractions involving cyclic mappings in non-Archimedean fuzzy metric spaces. Mathematics, 5.","DOI":"10.3390\/math5020022"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Saha, P., Guria, S., and Choudhury, B.S. (2019). Determining fuzzy distance through non-self fuzzy contractions. Yugoslav J. Oper. Res.","DOI":"10.2298\/YJOR180515002S"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"177","DOI":"10.5269\/bspm.v35i2.29466","article-title":"Some results on common best proximity point in fuzzy metric spaces","volume":"35","author":"Shayanpour","year":"2017","journal-title":"Bol. Soc. Paran. Mat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s12543-013-0155-z","article-title":"Best proximity point results in non-Archimedean fuzzy metric spaces","volume":"4","author":"Vetro","year":"2013","journal-title":"Fuzzy Inf. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2010). Fuzzy Mathematics: Approximation Theory, Springer.","DOI":"10.1007\/978-3-642-11220-1"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.fss.2011.12.008","article-title":"Some questions in fuzzy metric spaces","volume":"204","author":"Gregori","year":"2012","journal-title":"Fuzzy Sets Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/978-3-0348-8910-0_2","article-title":"Principle of weakly contractive maps in Hilbert spaces","volume":"98","author":"Alber","year":"1997","journal-title":"New Results Oper. Theory Its Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2683","DOI":"10.1016\/S0362-546X(01)00388-1","article-title":"Some theorems on weakly contractive maps","volume":"47","author":"Rhoades","year":"2001","journal-title":"Nonlinear Anal. TMA"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2116","DOI":"10.1016\/j.na.2010.11.017","article-title":"Fixed point theorems for generalized weakly contractive mappings","volume":"74","author":"Choudhury","year":"2011","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1896","DOI":"10.1016\/j.aml.2009.08.001","article-title":"Common fixed point for generalized (\u03c8,\u03d5)-weak contractions","volume":"22","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.aml.2008.02.007","article-title":"Fixed point theory for generalized \u03d5-weak contractions","volume":"22","author":"Zhang","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_31","unstructured":"Saha, P., Choudhury, B.S., and Das, P. A weak contraction in a fuzzy metric spaces, J. Uncertain Syst., in press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.fss.2003.09.007","article-title":"The Hausdorff fuzzy metric on compact sets","volume":"147","author":"Romaguera","year":"2004","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3787","DOI":"10.22436\/jnsa.009.06.28","article-title":"Optimal coincidence points of proximal quasi-contraction mappings in non-Archimedean fuzzy metric spaces","volume":"9","author":"Raza","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"6753","DOI":"10.2298\/FIL1819753K","article-title":"Best proximity coincidence point theorems for generalized nonlinear, contraction mappings","volume":"32","author":"Komal","year":"2018","journal-title":"Filomat"},{"key":"ref_35","unstructured":"Boothby, W. (1975). An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press."},{"key":"ref_36","unstructured":"Goldstein, H. (1951). Classical Mechanics, Addison-Wesley."},{"key":"ref_37","unstructured":"Chiang, A.C. (1992). Elements of Dynamic Optimization, McGraw-Hill."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/812\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:59:45Z","timestamp":1760187585000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/812"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,19]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060812"],"URL":"https:\/\/doi.org\/10.3390\/sym11060812","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,19]]}}}