{"id":"https://openalex.org/W4413951212","doi":"https://doi.org/10.3390/axioms14090679","title":"Oettli-Th\u00e9ra Theorem and Ekeland Variational Principle in Fuzzy b-Metric Spaces","display_name":"Oettli-Th\u00e9ra Theorem and Ekeland Variational Principle in Fuzzy b-Metric Spaces","publication_year":2025,"publication_date":"2025-09-03","ids":{"openalex":"https://openalex.org/W4413951212","doi":"https://doi.org/10.3390/axioms14090679"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14090679","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090679","pdf_url":"https://www.mdpi.com/2075-1680/14/9/679/pdf?version=1756904602","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/2075-1680/14/9/679/pdf?version=1756904602","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111460419","display_name":"Xuan Liu","orcid":"https://orcid.org/0009-0001-8219-6400"},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuan Liu","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066025506","display_name":"Fei He","orcid":"https://orcid.org/0000-0001-5638-0004"},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Fei He","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":"https://orcid.org/0000-0001-5638-0004","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035453504","display_name":"Ning Lu","orcid":"https://orcid.org/0000-0002-0496-7564"},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Ning Lu","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5035453504","https://openalex.org/A5066025506"],"corresponding_institution_ids":["https://openalex.org/I2722730"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.29778072,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"9","first_page":"679","last_page":"679"},"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"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9007999897003174,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/variational-principle","display_name":"Variational principle","score":0.6618199944496155},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6224097609519958},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.48721617460250854},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.469817191362381},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.41028711199760437},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3961144983768463},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.35201141238212585},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3308665454387665},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18935325741767883},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.1407153308391571},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07745176553726196},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.06551995873451233}],"concepts":[{"id":"https://openalex.org/C9259290","wikidata":"https://www.wikidata.org/wiki/Q745215","display_name":"Variational principle","level":2,"score":0.6618199944496155},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6224097609519958},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.48721617460250854},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.469817191362381},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41028711199760437},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3961144983768463},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.35201141238212585},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3308665454387665},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18935325741767883},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.1407153308391571},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07745176553726196},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.06551995873451233},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14090679","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090679","pdf_url":"https://www.mdpi.com/2075-1680/14/9/679/pdf?version=1756904602","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:75ebc04bb76c4109bac55bbbd16f5406","is_oa":true,"landing_page_url":"https://doaj.org/article/75ebc04bb76c4109bac55bbbd16f5406","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":"Axioms, Vol 14, Iss 9, p 679 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14090679","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14090679","pdf_url":"https://www.mdpi.com/2075-1680/14/9/679/pdf?version=1756904602","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":[{"id":"https://openalex.org/G7642712001","display_name":null,"funder_award_id":"12061050","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4413951212.pdf","grobid_xml":"https://content.openalex.org/works/W4413951212.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W72134831","https://openalex.org/W638584147","https://openalex.org/W1575008287","https://openalex.org/W1596940357","https://openalex.org/W1974932953","https://openalex.org/W1977750299","https://openalex.org/W1986618495","https://openalex.org/W1987761358","https://openalex.org/W2012795412","https://openalex.org/W2032966383","https://openalex.org/W2036649723","https://openalex.org/W2065765809","https://openalex.org/W2095740654","https://openalex.org/W2099626429","https://openalex.org/W2256003424","https://openalex.org/W2333277120","https://openalex.org/W2341296913","https://openalex.org/W2585543624","https://openalex.org/W2613089626","https://openalex.org/W2675766859","https://openalex.org/W2772466363","https://openalex.org/W2921583112","https://openalex.org/W2981486064","https://openalex.org/W3008766008","https://openalex.org/W3031847950","https://openalex.org/W3139030910","https://openalex.org/W3192612532","https://openalex.org/W3207944928","https://openalex.org/W4206140001","https://openalex.org/W4296041562","https://openalex.org/W4388651653","https://openalex.org/W4401201394"],"related_works":["https://openalex.org/W3111665545","https://openalex.org/W1986276133","https://openalex.org/W2039845181","https://openalex.org/W2398868469","https://openalex.org/W4363675004","https://openalex.org/W2095137273","https://openalex.org/W2119311694","https://openalex.org/W2001114394","https://openalex.org/W2255258360","https://openalex.org/W2107531168"],"abstract_inverted_index":{"The":[0],"purpose":[1],"of":[2,14,51,80,98],"this":[3,73],"paper":[4,74],"is":[5],"to":[6,94],"establish":[7],"the":[8,12,29,32,38,43,49,69,78,81,91,95],"Oettli\u2013The\u00b4ra":[9,44],"theorem":[10,63],"in":[11,64,72,84],"setting":[13],"KM-type":[15],"fuzzy":[16,65,86,99],"b-metric":[17,66,100],"spaces.":[18,67,101],"To":[19],"achieve":[20],"this,":[21],"we":[22,46],"first":[23],"prove":[24],"a":[25],"lemma":[26],"that":[27],"removes":[28],"constraints":[30],"on":[31,42],"space":[33],"coefficients,":[34],"which":[35],"significantly":[36],"simplifies":[37],"proof":[39],"process.":[40],"Based":[41],"theorem,":[45,58],"further":[47],"demonstrate":[48],"equivalence":[50],"Ekeland":[52],"variational":[53],"principle,":[54],"Caristi\u2019s":[55],"fixed":[56],"point":[57],"and":[59],"Takahashi\u2019s":[60],"nonconvex":[61],"minimization":[62],"Notably,":[68],"results":[70],"obtained":[71],"are":[75],"consistent":[76],"with":[77],"conditions":[79],"corresponding":[82],"theorems":[83],"classical":[85],"metric":[87],"spaces,":[88],"thereby":[89],"extending":[90],"existing":[92],"theories":[93],"broader":[96],"framework":[97]},"counts_by_year":[],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-09-04T00:00:00"}