{"id":"https://openalex.org/W2952535931","doi":"https://doi.org/10.3390/sym11060812","title":"Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions","display_name":"Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions","publication_year":2019,"publication_date":"2019-06-19","ids":{"openalex":"https://openalex.org/W2952535931","doi":"https://doi.org/10.3390/sym11060812","mag":"2952535931"},"language":"en","primary_location":{"id":"doi:10.3390/sym11060812","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060812","pdf_url":"https://www.mdpi.com/2073-8994/11/6/812/pdf?version=1561523367","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/6/812/pdf?version=1561523367","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045175955","display_name":"Parbati Saha","orcid":"https://orcid.org/0000-0003-0886-5386"},"institutions":[{"id":"https://openalex.org/I98365261","display_name":"Indian Institute of Engineering Science and Technology, Shibpur","ror":"https://ror.org/02ytfzr55","country_code":"IN","type":"education","lineage":["https://openalex.org/I98365261"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Parbati Saha","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India","institution_ids":["https://openalex.org/I98365261"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Shantau Guria","orcid":null},"institutions":[{"id":"https://openalex.org/I98365261","display_name":"Indian Institute of Engineering Science and Technology, Shibpur","ror":"https://ror.org/02ytfzr55","country_code":"IN","type":"education","lineage":["https://openalex.org/I98365261"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Shantau Guria","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India","institution_ids":["https://openalex.org/I98365261"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048600348","display_name":"Binayak S. Choudhury","orcid":"https://orcid.org/0000-0002-7263-6636"},"institutions":[{"id":"https://openalex.org/I98365261","display_name":"Indian Institute of Engineering Science and Technology, Shibpur","ror":"https://ror.org/02ytfzr55","country_code":"IN","type":"education","lineage":["https://openalex.org/I98365261"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Binayak S. Choudhury","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India","institution_ids":["https://openalex.org/I98365261"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042326806","display_name":"Manuel De la Sen","orcid":"https://orcid.org/0000-0001-9320-9433"},"institutions":[{"id":"https://openalex.org/I169108374","display_name":"University of the Basque Country","ror":"https://ror.org/000xsnr85","country_code":"ES","type":"education","lineage":["https://openalex.org/I169108374"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Manuel De la Sen","raw_affiliation_strings":["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"],"raw_orcid":"https://orcid.org/0000-0001-9320-9433","affiliations":[{"raw_affiliation_string":"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","institution_ids":["https://openalex.org/I169108374"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5042326806"],"corresponding_institution_ids":["https://openalex.org/I169108374"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3797,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.57225393,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"11","issue":"6","first_page":"812","last_page":"812"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7563974857330322},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6178454756736755},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5805575847625732},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.5774704217910767},{"id":"https://openalex.org/keywords/contraction","display_name":"Contraction (grammar)","score":0.5284057855606079},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.5009863376617432},{"id":"https://openalex.org/keywords/contraction-mapping","display_name":"Contraction mapping","score":0.4974208176136017},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.47320979833602905},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.4731360971927643},{"id":"https://openalex.org/keywords/fuzzy-subalgebra","display_name":"Fuzzy subalgebra","score":0.47083213925361633},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.4561462104320526},{"id":"https://openalex.org/keywords/signed-distance-function","display_name":"Signed distance function","score":0.41616183519363403},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.36374181509017944},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3609856367111206},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35545453429222107},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.35013705492019653},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.301349014043808},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.22788456082344055},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19179141521453857},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09650835394859314}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7563974857330322},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6178454756736755},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5805575847625732},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.5774704217910767},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.5284057855606079},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.5009863376617432},{"id":"https://openalex.org/C55524764","wikidata":"https://www.wikidata.org/wiki/Q515173","display_name":"Contraction mapping","level":3,"score":0.4974208176136017},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.47320979833602905},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.4731360971927643},{"id":"https://openalex.org/C115740575","wikidata":"https://www.wikidata.org/wiki/Q5511136","display_name":"Fuzzy subalgebra","level":5,"score":0.47083213925361633},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.4561462104320526},{"id":"https://openalex.org/C71169176","wikidata":"https://www.wikidata.org/wiki/Q7512907","display_name":"Signed distance function","level":2,"score":0.41616183519363403},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.36374181509017944},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3609856367111206},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35545453429222107},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.35013705492019653},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.301349014043808},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.22788456082344055},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19179141521453857},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09650835394859314},{"id":"https://openalex.org/C126322002","wikidata":"https://www.wikidata.org/wiki/Q11180","display_name":"Internal medicine","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym11060812","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060812","pdf_url":"https://www.mdpi.com/2073-8994/11/6/812/pdf?version=1561523367","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:9580984799564f5d835d5f794a135f1d","is_oa":true,"landing_page_url":"https://doaj.org/article/9580984799564f5d835d5f794a135f1d","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 11, Iss 6, p 812 (2019)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym11060812","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060812","pdf_url":"https://www.mdpi.com/2073-8994/11/6/812/pdf?version=1561523367","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W1245420","https://openalex.org/W164348063","https://openalex.org/W1487976166","https://openalex.org/W1566941664","https://openalex.org/W1577514385","https://openalex.org/W1596940357","https://openalex.org/W1968392362","https://openalex.org/W1977620557","https://openalex.org/W1981992983","https://openalex.org/W1987761358","https://openalex.org/W1998197580","https://openalex.org/W2000022332","https://openalex.org/W2005433363","https://openalex.org/W2020342106","https://openalex.org/W2026892575","https://openalex.org/W2041347874","https://openalex.org/W2042978270","https://openalex.org/W2074766818","https://openalex.org/W2074773690","https://openalex.org/W2077012380","https://openalex.org/W2084692626","https://openalex.org/W2099626429","https://openalex.org/W2103510752","https://openalex.org/W2134682153","https://openalex.org/W2238723887","https://openalex.org/W2327820785","https://openalex.org/W2478302134","https://openalex.org/W2552570644","https://openalex.org/W2586114308","https://openalex.org/W2604583505","https://openalex.org/W2912565176","https://openalex.org/W2917866716","https://openalex.org/W2940538715","https://openalex.org/W2954955438","https://openalex.org/W4211007335","https://openalex.org/W4285719527","https://openalex.org/W6690220121","https://openalex.org/W6730084316","https://openalex.org/W6765173123"],"related_works":["https://openalex.org/W2037428035","https://openalex.org/W2121560876","https://openalex.org/W2366798457","https://openalex.org/W2048452567","https://openalex.org/W4387806793","https://openalex.org/W2116600453","https://openalex.org/W2092290989","https://openalex.org/W2803916279","https://openalex.org/W1561699096","https://openalex.org/W2364431489"],"abstract_inverted_index":{"In":[0],"the":[1,6,10,23,29,48,101,127,138],"present":[2],"paper,":[3],"we":[4],"solve":[5],"problem":[7,24,30,50],"of":[8,16,31,37,89,113,140,146],"determining":[9],"fuzzy":[11,18,63,80,84,141],"distance":[12],"between":[13],"two":[14,110],"subsets":[15],"a":[17,38,73,79,83,117],"metric":[19,52],"space.":[20],"We":[21,61,104],"address":[22],"by":[25,144],"reducing":[26],"it":[27],"to":[28,78,95],"finding":[32],"an":[33,106],"optimal":[34],"approximate":[35],"solution":[36],"fixed":[39,119,147],"point":[40,59,120,148],"equation.":[41],"This":[42],"approach":[43],"is":[44,55,72,136],"well":[45],"studied":[46],"for":[47,66],"corresponding":[49],"in":[51,100,137],"spaces":[53,97],"and":[54,82,109],"known":[56],"as":[57],"proximity":[58],"problem.":[60],"employ":[62],"weak":[64,70],"contractions":[65],"that":[67,126],"purpose.":[68],"Fuzzy":[69,87],"contraction":[71,81],"recently":[74],"introduced":[75],"concept":[76],"intermediate":[77],"non-expansive":[85],"mapping.":[86],"versions":[88],"some":[90],"geometric":[91],"properties":[92],"essentially":[93],"belonging":[94],"Hilbert":[96],"are":[98],"considered":[99],"main":[102,128],"theorem.":[103,121],"include":[105],"illustrative":[107,123],"example":[108,124],"corollaries,":[111],"one":[112],"which":[114],"comes":[115],"from":[116],"well-known":[118],"The":[122,134],"shows":[125],"theorem":[129],"properly":[130],"includes":[131],"its":[132],"corollaries.":[133],"work":[135],"domain":[139],"global":[142],"optimization":[143],"use":[145],"methods.":[149]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}