{"id":"https://openalex.org/W4406797407","doi":"https://doi.org/10.3390/sym17020180","title":"Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces","display_name":"Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces","publication_year":2025,"publication_date":"2025-01-24","ids":{"openalex":"https://openalex.org/W4406797407","doi":"https://doi.org/10.3390/sym17020180"},"language":"en","primary_location":{"id":"doi:10.3390/sym17020180","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020180","pdf_url":"https://www.mdpi.com/2073-8994/17/2/180/pdf?version=1737729633","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/17/2/180/pdf?version=1737729633","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045304341","display_name":"Umar Ishtiaq","orcid":"https://orcid.org/0000-0002-5228-1073"},"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":"Umar Ishtiaq","raw_affiliation_strings":["Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan"],"affiliations":[{"raw_affiliation_string":"Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan","institution_ids":["https://openalex.org/I87482320"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056066746","display_name":"Fahad Jahangeer","orcid":null},"institutions":[{"id":"https://openalex.org/I121243025","display_name":"International Islamic University, Islamabad","ror":"https://ror.org/047w75g40","country_code":"PK","type":"education","lineage":["https://openalex.org/I121243025"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Fahad Jahangeer","raw_affiliation_strings":["Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan","institution_ids":["https://openalex.org/I121243025"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055287879","display_name":"M\u00fcbariz Garayev","orcid":"https://orcid.org/0000-0002-1715-7165"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Mubariz Garayev","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007601852","display_name":"Ioan\u2010Lucian Popa","orcid":"https://orcid.org/0000-0001-8042-1806"},"institutions":[{"id":"https://openalex.org/I3123573182","display_name":"Transylvania University of Bra\u0219ov","ror":"https://ror.org/01cg9ws23","country_code":"RO","type":"education","lineage":["https://openalex.org/I3123573182"]},{"id":"https://openalex.org/I4210140223","display_name":"1 Decembrie 1918 University","ror":"https://ror.org/04577k168","country_code":"RO","type":"education","lineage":["https://openalex.org/I4210140223"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Ioan-Lucian Popa","raw_affiliation_strings":["Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania","Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"],"affiliations":[{"raw_affiliation_string":"Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania","institution_ids":["https://openalex.org/I4210140223"]},{"raw_affiliation_string":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania","institution_ids":["https://openalex.org/I3123573182"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5007601852","https://openalex.org/A5045304341"],"corresponding_institution_ids":["https://openalex.org/I3123573182","https://openalex.org/I4210140223","https://openalex.org/I87482320"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":7.8292,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.95483374,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"17","issue":"2","first_page":"180","last_page":"180"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9994999766349792,"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":0.9994999766349792,"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/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9199000000953674,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7045773863792419},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.565131425857544},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.4554157257080078},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.4484706223011017},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.4394226670265198},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39600682258605957},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.3583958148956299},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3517271876335144},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3149591386318207},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23469668626785278},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.14017915725708008},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.11275053024291992}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7045773863792419},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.565131425857544},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4554157257080078},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.4484706223011017},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.4394226670265198},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39600682258605957},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3583958148956299},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3517271876335144},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3149591386318207},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23469668626785278},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.14017915725708008},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.11275053024291992},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17020180","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020180","pdf_url":"https://www.mdpi.com/2073-8994/17/2/180/pdf?version=1737729633","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"}],"best_oa_location":{"id":"doi:10.3390/sym17020180","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17020180","pdf_url":"https://www.mdpi.com/2073-8994/17/2/180/pdf?version=1737729633","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":[{"id":"https://openalex.org/G1198819986","display_name":null,"funder_award_id":"RSPD2025R1056","funder_id":"https://openalex.org/F4320321145","funder_display_name":"King Saud University"},{"id":"https://openalex.org/G8239988234","display_name":null,"funder_award_id":"Riyadh","funder_id":"https://openalex.org/F4320321145","funder_display_name":"King Saud University"}],"funders":[{"id":"https://openalex.org/F4320321145","display_name":"King Saud University","ror":"https://ror.org/02f81g417"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4406797407.pdf"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1458771408","https://openalex.org/W1968392362","https://openalex.org/W1987761358","https://openalex.org/W2000935779","https://openalex.org/W2042978270","https://openalex.org/W2079096866","https://openalex.org/W3004323094","https://openalex.org/W3004879613","https://openalex.org/W3103202097","https://openalex.org/W3180362899","https://openalex.org/W3202621566","https://openalex.org/W3215818013","https://openalex.org/W4206488922","https://openalex.org/W4211007335","https://openalex.org/W4377833827","https://openalex.org/W4380300907","https://openalex.org/W4385341742","https://openalex.org/W4386336842","https://openalex.org/W4386525235","https://openalex.org/W4387781612","https://openalex.org/W4390693872","https://openalex.org/W4391575869","https://openalex.org/W4399906831","https://openalex.org/W4400019031","https://openalex.org/W4402371148","https://openalex.org/W6635703843","https://openalex.org/W6642368997","https://openalex.org/W6735707003","https://openalex.org/W6758581921","https://openalex.org/W6805036799","https://openalex.org/W6852529156"],"related_works":["https://openalex.org/W571281153","https://openalex.org/W2949403936","https://openalex.org/W4301247218","https://openalex.org/W3023806604","https://openalex.org/W2952139525","https://openalex.org/W2129818458","https://openalex.org/W4289551052","https://openalex.org/W2340777379","https://openalex.org/W4390756076","https://openalex.org/W4243074410"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,54,80,99],"present":[4,100],"the":[5,10,27,45,49,87,90,94,97,105,118],"concept":[6],"of":[7,12,44,109,112],"(\u03a5,\u03a9)-iterativemappings":[8],"in":[9,20,41,47,93],"setting":[11],"fuzzy":[13,21],"bipolar":[14,22,64],"metric":[15,23],"space.":[16],"The":[17],"symmetric":[18],"property":[19],"spaces":[24],"guarantees":[25],"that":[26],"distance":[28],"between":[29],"any":[30],"two":[31],"elements":[32,50],"remains":[33],"invariant":[34],"under":[35],"permutation,":[36],"ensuring":[37],"consistency":[38],"and":[39,75,107],"uniformity":[40],"measurement":[42],"regardless":[43],"order":[46],"which":[48],"are":[51],"considered.":[52],"Furthermore,":[53,79],"prove":[55],"several":[56],"best":[57],"proximity":[58],"point":[59],"results":[60,92],"by":[61,116],"utilizing":[62],"(\u03a5,\u03a9)-fuzzy":[63],"proximal":[65,69,73],"contraction,":[66,70],"(\u03a5,\u03a9)-Reich\u2013Rus\u2013Ciric":[67],"type":[68,72,77],"(\u03a5,\u03a9)-Kannan":[71],"contraction":[74],"(\u03a5,\u03a9)-Hardy\u2013Rogers":[76],"contraction.":[78],"provide":[81],"some":[82],"non-trivial":[83],"examples":[84],"to":[85,103],"show":[86],"comparison":[88],"with":[89],"existing":[91],"literature.":[95],"At":[96],"end,":[98],"an":[101,113],"application":[102],"find":[104],"existence":[106],"uniqueness":[108],"a":[110],"solution":[111],"integral":[114],"equation":[115],"applying":[117],"main":[119],"result.":[120]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}