{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T12:39:52Z","timestamp":1761395992716,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,3,29]],"date-time":"2021-03-29T00:00:00Z","timestamp":1616976000000},"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 setting of fuzzy metric spaces (FMSs), a global optimization problem (GOP) obtaining the distance between two subsets of an FMS is solved by a tripled fixed-point (FP) technique here. Also, fuzzy weak tripled contractions (WTCs) for that are given. This problem was known before in metric space (MS) as a proximity point problem (PPP). The result is correct for each continuous \u03c4\u2014norms related to the FMS. Furthermore, a non-trivial example to illustrate the main theorem is discussed.<\/jats:p>","DOI":"10.3390\/sym13040565","type":"journal-article","created":{"date-parts":[[2021,3,29]],"date-time":"2021-03-29T16:01:57Z","timestamp":1617033717000},"page":"565","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Weak Tripled Contraction for Solving a Fuzzy Global Optimization Problem in Fuzzy Metric Spaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-9367","authenticated-orcid":false,"given":"Hasanen A.","family":"Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"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, University of the Basque Country, 48940 Leioa, Bizkaia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.jmaa.2005.10.081","article-title":"Existence and convergence of best proximity points","volume":"323","author":"Eldred","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3790","DOI":"10.1016\/j.na.2007.10.014","article-title":"Best proximity points for cyclic Meir\u2013Keeler contractions","volume":"69","author":"Bari","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s10013-015-0141-3","article-title":"Best proximity point results in generalized metric spaces","volume":"44","author":"Choudhury","year":"2016","journal-title":"Vietnam J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1016\/j.bulsci.2013.02.003","article-title":"Best proximity points for \u03b1\u2013\u03c8\u2014Proximal contractive type mappings and applications","volume":"137","author":"Jleli","year":"2013","journal-title":"Bull. Sci. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jleli, M., Karapinar, K., and Samet, B. (2012). Best proximity point result for MK-proximal contractions. Abstr. Appl. Anal.","DOI":"10.1155\/2012\/193085"},{"key":"ref_6","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_7","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_8","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_9","doi-asserted-by":"crossref","unstructured":"Sintunavarat, W., and Kumam, P. (2012). Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl.","DOI":"10.1186\/1687-1812-2012-93"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.amc.2016.06.022","article-title":"Error estimates for approximation of coupled best proximity points for cyclic contractive maps","volume":"290","author":"Ilchev","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.mcm.2011.01.036","article-title":"Coupled fixed point results in generalized metric spaces","volume":"54","author":"Choudhury","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_12","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_13","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_14","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_15","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.chaos.2008.11.010","article-title":"Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces","volume":"42","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s40574-015-0044-y","article-title":"A new contractive mapping principle in fuzzy metric spaces","volume":"8","author":"Saha","year":"2016","journal-title":"Bull. Univ. Math. Ital."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Saha, P., Guria, S., and Choudhury, B.S. (2019). Determining fuzzy distance through non-self fuzzy contractions. Yugosl. J. Oper. Res.","DOI":"10.2298\/YJOR180515002S"},{"key":"ref_19","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. Math."},{"key":"ref_20","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":"5","author":"Vetro","year":"2013","journal-title":"Fuzzy Inf. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s13398-012-0095-1","article-title":"Two coupled weak contraction theorems in partially ordered metric spaces","volume":"108","author":"Choudhury","year":"2014","journal-title":"RACSAM"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.fiae.2016.06.005","article-title":"Weak coupled coincidence point results having a partially ordering in fuzzy metric spaces","volume":"8","author":"Saha","year":"2016","journal-title":"Fuzzy Inf. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/0362-546X(87)90077-0","article-title":"Coupled fixed points of nonlinear operators with applications","volume":"11","author":"Guo","year":"1987","journal-title":"Nonlinear Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3656","DOI":"10.1016\/j.camwa.2010.03.062","article-title":"Coupled fixed point theorems for nonlinear contractions in cone metric spaces","volume":"59","author":"Karapinar","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1016\/j.na.2010.09.055","article-title":"Coupled fixed points in partially ordered metric spaces and application","volume":"74","author":"Luong","year":"2011","journal-title":"Nonlinear Anal."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hammad, H.A., and la Sen, M.D. (2019). A coupled fixed point technique for solving coupled systems of functional and nonlinear integral equations. Mathematics, 7.","DOI":"10.3390\/math7070634"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1186\/s42787-019-0064-3","article-title":"Coupled coincidence point technique and its application for solving nonlinear integral equations in RPOCbML spaces","volume":"28","author":"Hammad","year":"2020","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_28","first-page":"7","article-title":"Principle of weakly contractive maps in Hilbert spaces","volume":"Volume 98","author":"Alber","year":"1997","journal-title":"Operator Theory: Advances and Applications"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.aml.2008.02.007","article-title":"Fixed point theory for generalized \u03d5\u2014Weak contractions","volume":"22","author":"Zhang","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cho, Y.J., Kadelburg, Z., Saadati, R., and Shatanawi, W. (2012). Coupled fixed point theorems under weak contractions. Discrete Dyn. Nat. Soc.","DOI":"10.1155\/2012\/184534"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s00574-019-00144-1","article-title":"Solution of nonlinear integral equation via fixed point of cyclic \u03b1L\u03c8\u2014Rational contraction mappings in metric-like spaces","volume":"51","author":"Hammad","year":"2020","journal-title":"Bull. Braz. Math. Soc. New Ser."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hammad, H.A., and la Sen, M.D. (2019). Generalized contractive mappings and related results in b-metric-like spaces with an application. Symmetry, 11.","DOI":"10.3390\/sym11050667"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4889","DOI":"10.1016\/j.na.2011.03.032","article-title":"Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces","volume":"74","author":"Berinde","year":"2011","journal-title":"Nonlinear Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5929","DOI":"10.1016\/j.amc.2011.11.049","article-title":"Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces","volume":"218","author":"Borcut","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1186\/1687-1812-2012-134","article-title":"Tripled fixed point of W-compatible mappings in abstract metric spaces","volume":"2012","author":"Aydi","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1186\/1029-242X-2013-453","article-title":"Existence of a tripled coincidence point in ordered Gb-metric spaces and applications to a system of integral equations","volume":"2013","author":"Mustafa","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.amc.2014.03.059","article-title":"A note on tripled coincidence and tripled common fixed point theorems in partially ordered metric spaces","volume":"236","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1186\/s13660-020-02477-8","article-title":"A technique of tripled coincidence points for solving a system of nonlinear integral equations in POCML spaces","volume":"2020","author":"Hammad","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1186\/s13662-020-03023-y","article-title":"A tripled fixed point technique for solving a tripled-system of integral equations and Markov process in CCbMS","volume":"2020","author":"Hammad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_40","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. Math."},{"key":"ref_41","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":"Grabice","year":"1988","journal-title":"Fuzzy Sets Syst."},{"key":"ref_42","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":"Ramaguera","year":"2004","journal-title":"Fuzzy Sets Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/565\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:10:19Z","timestamp":1760364619000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/565"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,29]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["sym13040565"],"URL":"https:\/\/doi.org\/10.3390\/sym13040565","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,3,29]]}}}