{"id":"https://openalex.org/W3014594096","doi":"https://doi.org/10.3390/axioms9020036","title":"Fuzzy b-Metric Spaces: Fixed Point Results for \u03c8-Contraction Correspondences and Their Application","display_name":"Fuzzy b-Metric Spaces: Fixed Point Results for \u03c8-Contraction Correspondences and Their Application","publication_year":2020,"publication_date":"2020-03-31","ids":{"openalex":"https://openalex.org/W3014594096","doi":"https://doi.org/10.3390/axioms9020036","mag":"3014594096"},"language":"en","primary_location":{"id":"doi:10.3390/axioms9020036","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms9020036","pdf_url":"https://www.mdpi.com/2075-1680/9/2/36/pdf?version=1587138076","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/9/2/36/pdf?version=1587138076","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054055136","display_name":"Mujahid Abbas","orcid":"https://orcid.org/0000-0001-5528-1207"},"institutions":[{"id":"https://openalex.org/I43879129","display_name":"Government College University, Lahore","ror":"https://ror.org/040gec961","country_code":"PK","type":"education","lineage":["https://openalex.org/I43879129"]},{"id":"https://openalex.org/I69552723","display_name":"University of Pretoria","ror":"https://ror.org/00g0p6g84","country_code":"ZA","type":"education","lineage":["https://openalex.org/I69552723"]}],"countries":["PK","ZA"],"is_corresponding":false,"raw_author_name":"Mujahid Abbas","raw_affiliation_strings":["Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa","Department of Mathematics, Government College University, Lahore 54770, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa","institution_ids":["https://openalex.org/I69552723"]},{"raw_affiliation_string":"Department of Mathematics, Government College University, Lahore 54770, Pakistan","institution_ids":["https://openalex.org/I43879129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053287765","display_name":"Fatemeh Lael","orcid":"https://orcid.org/0000-0003-3665-3401"},"institutions":[{"id":"https://openalex.org/I3131771360","display_name":"Buein Zahra Technical University","ror":"https://ror.org/02nee2y72","country_code":"IR","type":"education","lineage":["https://openalex.org/I3131771360"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Fatemeh Lael","raw_affiliation_strings":["Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin 3451745346, Iran"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin 3451745346, Iran","institution_ids":["https://openalex.org/I3131771360"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064449182","display_name":"Naeem Saleem","orcid":"https://orcid.org/0000-0002-1485-6163"},"institutions":[{"id":"https://openalex.org/I87482320","display_name":"University of Management and Technology","ror":"https://ror.org/0095xcq10","country_code":"PK","type":"education","lineage":["https://openalex.org/I87482320"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Naeem Saleem","raw_affiliation_strings":["Department of Mathematics, University of Management and Technology, Lahore 54400, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Management and Technology, Lahore 54400, Pakistan","institution_ids":["https://openalex.org/I87482320"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5064449182"],"corresponding_institution_ids":["https://openalex.org/I87482320"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":5.0618,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.95745568,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"9","issue":"2","first_page":"36","last_page":"36"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9998999834060669,"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":0.9998999834060669,"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":0.7383167147636414},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.6904243230819702},{"id":"https://openalex.org/keywords/contraction","display_name":"Contraction (grammar)","score":0.5591045618057251},{"id":"https://openalex.org/keywords/t-norm","display_name":"T-norm","score":0.5508831739425659},{"id":"https://openalex.org/keywords/metric-map","display_name":"Metric map","score":0.49656563997268677},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.4714314043521881},{"id":"https://openalex.org/keywords/usability","display_name":"Usability","score":0.4537111520767212},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4507914185523987},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4475293755531311},{"id":"https://openalex.org/keywords/convex-metric-space","display_name":"Convex metric space","score":0.44700151681900024},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.44420403242111206},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.4432269036769867},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.41759246587753296},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.30136609077453613},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.2887169420719147},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23445698618888855},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2264116108417511},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15042120218276978},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09299305081367493}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7383167147636414},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.6904243230819702},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.5591045618057251},{"id":"https://openalex.org/C2721081","wikidata":"https://www.wikidata.org/wiki/Q2383161","display_name":"T-norm","level":5,"score":0.5508831739425659},{"id":"https://openalex.org/C145084290","wikidata":"https://www.wikidata.org/wiki/Q2713824","display_name":"Metric map","level":4,"score":0.49656563997268677},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.4714314043521881},{"id":"https://openalex.org/C170130773","wikidata":"https://www.wikidata.org/wiki/Q216378","display_name":"Usability","level":2,"score":0.4537111520767212},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4507914185523987},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4475293755531311},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.44700151681900024},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.44420403242111206},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.4432269036769867},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.41759246587753296},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.30136609077453613},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.2887169420719147},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23445698618888855},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2264116108417511},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15042120218276978},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09299305081367493},{"id":"https://openalex.org/C107457646","wikidata":"https://www.wikidata.org/wiki/Q207434","display_name":"Human\u2013computer interaction","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},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C126322002","wikidata":"https://www.wikidata.org/wiki/Q11180","display_name":"Internal medicine","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/axioms9020036","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms9020036","pdf_url":"https://www.mdpi.com/2075-1680/9/2/36/pdf?version=1587138076","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:cdafb6442fa24cb78f6136e51a308bc0","is_oa":true,"landing_page_url":"https://doaj.org/article/cdafb6442fa24cb78f6136e51a308bc0","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 9, Iss 2, p 36 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/9/2/36/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms9020036","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 9; Issue 2; Pages: 36","raw_type":"Text"},{"id":"pmh:oai:repository.up.ac.za:2263/76461","is_oa":true,"landing_page_url":"http://hdl.handle.net/2263/76461","pdf_url":null,"source":{"id":"https://openalex.org/S4306401870","display_name":"UpSpace Institutional Repository (University of Pretoria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I69552723","host_organization_name":"University of Pretoria","host_organization_lineage":["https://openalex.org/I69552723"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.3390/axioms9020036","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms9020036","pdf_url":"https://www.mdpi.com/2075-1680/9/2/36/pdf?version=1587138076","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3014594096.pdf","grobid_xml":"https://content.openalex.org/works/W3014594096.grobid-xml"},"referenced_works_count":48,"referenced_works":["https://openalex.org/W72134831","https://openalex.org/W1458771408","https://openalex.org/W1596940357","https://openalex.org/W1853225040","https://openalex.org/W1966503202","https://openalex.org/W1968172834","https://openalex.org/W1977299782","https://openalex.org/W1981992983","https://openalex.org/W1987761358","https://openalex.org/W2003646949","https://openalex.org/W2004781734","https://openalex.org/W2010539741","https://openalex.org/W2022123893","https://openalex.org/W2040949060","https://openalex.org/W2048705177","https://openalex.org/W2061177395","https://openalex.org/W2063217364","https://openalex.org/W2074564900","https://openalex.org/W2074773690","https://openalex.org/W2079315231","https://openalex.org/W2079622135","https://openalex.org/W2080257591","https://openalex.org/W2095244856","https://openalex.org/W2129941846","https://openalex.org/W2146407970","https://openalex.org/W2156533025","https://openalex.org/W2164883554","https://openalex.org/W2317666921","https://openalex.org/W2435984206","https://openalex.org/W2500245806","https://openalex.org/W2503361663","https://openalex.org/W2586114308","https://openalex.org/W2587660114","https://openalex.org/W2594987052","https://openalex.org/W2604583505","https://openalex.org/W2782853315","https://openalex.org/W2904071456","https://openalex.org/W2913670109","https://openalex.org/W2963515256","https://openalex.org/W2969884762","https://openalex.org/W3098986449","https://openalex.org/W6602953246","https://openalex.org/W6666435611","https://openalex.org/W6683028690","https://openalex.org/W6718417300","https://openalex.org/W6733594437","https://openalex.org/W6759297742","https://openalex.org/W6767261595"],"related_works":["https://openalex.org/W2470367325","https://openalex.org/W2610752199","https://openalex.org/W1971871035","https://openalex.org/W2095857608","https://openalex.org/W1587207900","https://openalex.org/W1666773474","https://openalex.org/W4323049087","https://openalex.org/W4287753276","https://openalex.org/W2184304495","https://openalex.org/W3025992958"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,44,63],"introduce":[4],"the":[5,59,69,72],"concepts":[6],"of":[7,71],"\u03c8":[8,12],"-contraction":[9,13],"and":[10,19,39,52],"monotone":[11],"correspondence":[14],"in":[15,35,49],"\u201cfuzzy":[16,40,53],"b-metric":[17,41],"spaces\u201d":[18],"obtain":[20],"fixed":[21],"point":[22],"results":[23,30,60],"for":[24],"these":[25],"contractive":[26],"mappings.":[27],"The":[28],"obtained":[29,61,73],"generalize":[31],"some":[32],"existing":[33],"ones":[34],"fuzzy":[36],"metric":[37],"spaces":[38],"spaces\u201d.":[42,56],"Further":[43],"address":[45],"an":[46,65],"open":[47],"problem":[48],"b":[50,54],"-metric":[51,55],"To":[57],"elaborate":[58],"herein":[62],"provide":[64],"example":[66],"that":[67],"shows":[68],"usability":[70],"results.":[74]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
