{"id":"https://openalex.org/W7225337151","doi":"https://doi.org/10.3390/axioms15110826","title":"Extended Generalized Contractions, Metric Completeness, and Applications to Solving Nonlinear Diophantine Equations","display_name":"Extended Generalized Contractions, Metric Completeness, and Applications to Solving Nonlinear Diophantine Equations","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225337151","doi":"https://doi.org/10.3390/axioms15110826"},"language":"en","primary_location":{"id":"doi:10.3390/axioms15110826","doi":"https://doi.org/10.3390/axioms15110826","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms15110826","pdf_url":null,"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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/axioms15110826","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052436825","display_name":"Ravindra Kishor Bisht","orcid":"https://orcid.org/0000-0002-6333-8127","observed_orcids":["https://orcid.org/0000-0002-6333-8127"]},"institutions":[{"id":"https://openalex.org/I11414854","display_name":"National Defence Academy","ror":"https://ror.org/021twv479","country_code":"IN","type":"education","lineage":["https://openalex.org/I11414854"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Ravindra K. Bisht","raw_affiliation_strings":["Department of Mathematics, National Defence Academy, Khadakwasla, Pune 411023, India"],"raw_orcid":"https://orcid.org/0000-0002-6333-8127","affiliations":[{"raw_affiliation_string":"Department of Mathematics, National Defence Academy, Khadakwasla, Pune 411023, India","institution_ids":["https://openalex.org/I11414854"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042326806","display_name":"Manuel De la Sen","orcid":"https://orcid.org/0000-0001-9320-9433","observed_orcids":["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":false,"raw_author_name":"Manuel De La Sen","raw_affiliation_strings":["Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa, Spain"],"raw_orcid":"https://orcid.org/0000-0001-9320-9433","affiliations":[{"raw_affiliation_string":"Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa, Spain","institution_ids":["https://openalex.org/I169108374"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5052436825"],"corresponding_institution_ids":["https://openalex.org/I11414854"],"apc_list":{"value":2400,"currency":"CHF","value_usd":2896},"apc_paid":{"value":2400,"currency":"CHF","value_usd":2896},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"15","issue":"11","first_page":"826","last_page":"826"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":1.0,"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":1.0,"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/T12508","display_name":"Functional Equations Stability Results","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/generalized-contractions","display_name":"generalized contractions","score":0.8450000286102295},{"id":"https://openalex.org/keywords/fixed-point-theorems","display_name":"fixed point theorems","score":0.9100000262260437},{"id":"https://openalex.org/keywords/metric-completeness","display_name":"metric completeness","score":0.7440000176429749},{"id":"https://openalex.org/keywords/nonexpansive-mappings","display_name":"nonexpansive mappings","score":0.8180000185966492},{"id":"https://openalex.org/keywords/multiple-fixed-points","display_name":"multiple fixed points","score":0.8740000128746033},{"id":"https://openalex.org/keywords/discontinuous-mappings","display_name":"discontinuous maps","score":0.6340000033378601},{"id":"https://openalex.org/keywords/rational-contractions","display_name":"rational contractions","score":0.6570000052452087},{"id":"https://openalex.org/keywords/meir-keeler-contractions","display_name":"Meir-Keeler contractions","score":0.8069999814033508},{"id":"https://openalex.org/keywords/nonlinear-diophantine-equations","display_name":"nonlinear Diophantine equations","score":0.9789999723434448},{"id":"https://openalex.org/keywords/integer-solutions","display_name":"integer solutions","score":0.8100000023841858},{"id":"https://openalex.org/keywords/roots-of-unity","display_name":"roots of unity","score":0.718999981880188}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8751999735832214},{"id":"https://openalex.org/C206530611","wikidata":"https://www.wikidata.org/wiki/Q905896","display_name":"Diophantine equation","level":2,"score":0.6920999884605408},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.5115000009536743},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4778999984264374},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.45750001072883606},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.45249998569488525},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.45019999146461487},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.4487999975681305},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4440999925136566},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.44040000438690186},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3804999887943268},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.36230000853538513},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3540000021457672},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.3441999852657318},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3409000039100647},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32919999957084656},{"id":"https://openalex.org/C102966492","wikidata":"https://www.wikidata.org/wiki/Q1227061","display_name":"Diophantine approximation","level":3,"score":0.3203999996185303},{"id":"https://openalex.org/C555856410","wikidata":"https://www.wikidata.org/wiki/Q65122107","display_name":"Diophantine geometry","level":4,"score":0.31929999589920044},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.2946999967098236},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2921000123023987},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2831999957561493},{"id":"https://openalex.org/C203192437","wikidata":"https://www.wikidata.org/wiki/Q217847","display_name":"Cauchy sequence","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.26669999957084656},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.266400009393692},{"id":"https://openalex.org/C35767895","wikidata":"https://www.wikidata.org/wiki/Q5141583","display_name":"Coincidence point","level":3,"score":0.26010000705718994},{"id":"https://openalex.org/C138268822","wikidata":"https://www.wikidata.org/wiki/Q1051925","display_name":"Resolution (logic)","level":2,"score":0.2513999938964844}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/axioms15110826","doi":"https://doi.org/10.3390/axioms15110826","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms15110826","pdf_url":null,"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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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"}],"best_oa_location":{"id":"doi:10.3390/axioms15110826","doi":"https://doi.org/10.3390/axioms15110826","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms15110826","pdf_url":null,"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,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-2","vabb-shw"],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W147294779","https://openalex.org/W930817409","https://openalex.org/W1029025877","https://openalex.org/W1458771408","https://openalex.org/W1574920830","https://openalex.org/W1595749660","https://openalex.org/W1878797489","https://openalex.org/W1972565515","https://openalex.org/W1981459842","https://openalex.org/W1984781556","https://openalex.org/W2002573556","https://openalex.org/W2027735821","https://openalex.org/W2036451606","https://openalex.org/W2048884582","https://openalex.org/W2054618054","https://openalex.org/W2089814328","https://openalex.org/W2116659777","https://openalex.org/W2283642325","https://openalex.org/W2321163659","https://openalex.org/W2332549331","https://openalex.org/W2792580891","https://openalex.org/W2793435614","https://openalex.org/W2900322214","https://openalex.org/W2910729755","https://openalex.org/W2977925599","https://openalex.org/W2996919789","https://openalex.org/W3048168289","https://openalex.org/W3088062271","https://openalex.org/W3154212811","https://openalex.org/W4312126008","https://openalex.org/W4385843760","https://openalex.org/W4391541816","https://openalex.org/W4394620885","https://openalex.org/W4401764596","https://openalex.org/W4402824834","https://openalex.org/W4404837354","https://openalex.org/W4408924996","https://openalex.org/W4409063499","https://openalex.org/W4414467888","https://openalex.org/W7133296280","https://openalex.org/W7161005877"],"related_works":["https://openalex.org/W3090621888","https://openalex.org/W3208048653","https://openalex.org/W2594394105","https://openalex.org/W4386829592","https://openalex.org/W4392236741","https://openalex.org/W1616351379","https://openalex.org/W7223086743","https://openalex.org/W3185529753","https://openalex.org/W2975176681","https://openalex.org/W2155754226"],"abstract_inverted_index":{"In":[0],"this":[1,149],"paper,":[2],"we":[3],"introduce":[4],"a":[5,12,17,45,70,79,124,172,180],"generalized":[6],"(\u03b5-\u03b4)-type":[7],"contractive":[8,39],"condition":[9],"and":[10,40,43,67,147],"develop":[11],"unified":[13],"framework":[14,36],"that":[15,84,88,99],"extends":[16],"wide":[18],"spectrum":[19],"of":[20,48,54,74,81,89,138,160,171],"classical":[21,49],"fixed-point":[22,50],"results":[23,178],"for":[24],"self-mappings":[25],"in":[26,91,94,127,179],"complete":[27],"metric":[28,82],"spaces,":[29],"with":[30],"or":[31],"without":[32],"continuity":[33],"assumptions.":[34],"The":[35,114],"accommodates":[37],"both":[38],"nonexpansive":[41],"mappings":[42,98,109],"unifies":[44],"broad":[46],"class":[47],"theorems,":[51],"including":[52,123],"those":[53],"Banach,":[55],"Kannan,":[56],"Chatterjea,":[57],"Bianchini,":[58],"Matkowski,":[59],"Boyd\u2013Wong,":[60],"Meir\u2013Keeler,":[61],"Reich,":[62],"\u0106iri\u0107,":[63],"\u0106iri\u0107\u2013Jachymski\u2013Matkowski,":[64],"rational":[65],"contractions,":[66],"Pant,":[68],"within":[69],"multiple-fixed-point":[71],"setting.":[72,183],"One":[73],"the":[75,128,135,141,157,168,177],"established":[76],"theorems":[77],"yields":[78],"characterization":[80],"completeness":[83],"is":[85],"different":[86],"from":[87],"Subrahmanyam":[90],"his":[92],"research":[93],"1975,":[95],"which":[96],"covers":[97],"are":[100,117,134],"discontinuous":[101],"at":[102],"their":[103],"fixed":[104,112,132],"points":[105,133],"as":[106,108],"well":[107],"admitting":[110],"multiple":[111],"points.":[113],"theoretical":[115],"developments":[116],"illustrated":[118],"through":[119],"several":[120],"nontrivial":[121],"examples,":[122],"geometric":[125],"example":[126,166],"complex":[129],"plane":[130],"whose":[131],"Nth":[136],"roots":[137],"unity.":[139],"Following":[140],"initial":[142],"2024":[143],"resolution":[144],"by":[145],"Pant":[146],"collaborators,":[148],"paper":[150],"provides":[151],"an":[152],"alternative":[153],"constructive":[154],"solution":[155],"to":[156],"strong":[158],"version":[159],"Rhoades\u2019":[161],"1988":[162],"open":[163],"problem.":[164],"An":[165],"on":[167],"integer":[169],"solutions":[170],"nonlinear":[173],"Diophantine":[174],"equation":[175],"illustrates":[176],"discrete":[181],"arithmetic":[182]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}
