{"id":"https://openalex.org/W3154349298","doi":"https://doi.org/10.3390/axioms10020058","title":"Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov\u2013Razumikhin Method","display_name":"Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov\u2013Razumikhin Method","publication_year":2021,"publication_date":"2021-04-09","ids":{"openalex":"https://openalex.org/W3154349298","doi":"https://doi.org/10.3390/axioms10020058","mag":"3154349298"},"language":"en","primary_location":{"id":"doi:10.3390/axioms10020058","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10020058","pdf_url":"https://www.mdpi.com/2075-1680/10/2/58/pdf?version=1618461427","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/2075-1680/10/2/58/pdf?version=1618461427","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085174760","display_name":"Osman Tun\u00e7","orcid":"https://orcid.org/0000-0003-2965-4561"},"institutions":[{"id":"https://openalex.org/I118567650","display_name":"Van Y\u00fcz\u00fcnc\u00fc Y\u0131l \u00dcniversitesi","ror":"https://ror.org/041jyzp61","country_code":"TR","type":"education","lineage":["https://openalex.org/I118567650"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Osman Tun\u00e7","raw_affiliation_strings":["Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, 65080 Van, Turkey"],"raw_orcid":"https://orcid.org/0000-0003-2965-4561","affiliations":[{"raw_affiliation_string":"Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, 65080 Van, Turkey","institution_ids":["https://openalex.org/I118567650"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081659070","display_name":"\u00d6zkan Atan","orcid":"https://orcid.org/0000-0001-6443-9600"},"institutions":[{"id":"https://openalex.org/I118567650","display_name":"Van Y\u00fcz\u00fcnc\u00fc Y\u0131l \u00dcniversitesi","ror":"https://ror.org/041jyzp61","country_code":"TR","type":"education","lineage":["https://openalex.org/I118567650"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"\u00d6zkan Atan","raw_affiliation_strings":["Department of Electric-Electronic Engineering, Faculty of Engineering, Van Yuzuncu Yil University, 65080 Van, Turkey"],"raw_orcid":"https://orcid.org/0000-0001-6443-9600","affiliations":[{"raw_affiliation_string":"Department of Electric-Electronic Engineering, Faculty of Engineering, Van Yuzuncu Yil University, 65080 Van, Turkey","institution_ids":["https://openalex.org/I118567650"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054651401","display_name":"Cemil Tun\u00e7","orcid":"https://orcid.org/0000-0003-2909-8753"},"institutions":[{"id":"https://openalex.org/I118567650","display_name":"Van Y\u00fcz\u00fcnc\u00fc Y\u0131l \u00dcniversitesi","ror":"https://ror.org/041jyzp61","country_code":"TR","type":"education","lineage":["https://openalex.org/I118567650"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Cemil Tun\u00e7","raw_affiliation_strings":["Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey"],"raw_orcid":"https://orcid.org/0000-0003-2909-8753","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey","institution_ids":["https://openalex.org/I118567650"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050520268","display_name":"Jen\u2010Chih Yao","orcid":"https://orcid.org/0000-0001-9041-3375"},"institutions":[{"id":"https://openalex.org/I184693016","display_name":"China Medical University","ror":"https://ror.org/00v408z34","country_code":"TW","type":"education","lineage":["https://openalex.org/I184693016"]},{"id":"https://openalex.org/I4210126829","display_name":"China Medical University Hospital","ror":"https://ror.org/0368s4g32","country_code":"TW","type":"healthcare","lineage":["https://openalex.org/I4210126829"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Jen-Chih Yao","raw_affiliation_strings":["Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 406, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 406, Taiwan","institution_ids":["https://openalex.org/I4210126829","https://openalex.org/I184693016"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5050520268"],"corresponding_institution_ids":["https://openalex.org/I184693016","https://openalex.org/I4210126829"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":3.0116,"has_fulltext":true,"cited_by_count":33,"citation_normalized_percentile":{"value":0.91854063,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"10","issue":"2","first_page":"58","last_page":"58"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T11081","display_name":"Advanced Control Systems Design","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9988999962806702,"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/mathematics","display_name":"Mathematics","score":0.7847001552581787},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.7306103706359863},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.7264301776885986},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6562298536300659},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.562584638595581},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"Exponential stability","score":0.5307111144065857},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5023555755615234},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5017127990722656},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4369059205055237},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.40932849049568176},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10921430587768555},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.040254682302474976}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7847001552581787},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.7306103706359863},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.7264301776885986},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6562298536300659},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.562584638595581},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.5307111144065857},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5023555755615234},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5017127990722656},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4369059205055237},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40932849049568176},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10921430587768555},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.040254682302474976},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/axioms10020058","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10020058","pdf_url":"https://www.mdpi.com/2075-1680/10/2/58/pdf?version=1618461427","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:31dc97e3-8c0e-4c84-83fd-f6b1f5a1aeef","is_oa":false,"landing_page_url":"https://avesis.yyu.edu.tr/publication/details/31dc97e3-8c0e-4c84-83fd-f6b1f5a1aeef/oai","pdf_url":null,"source":{"id":"https://openalex.org/S7407055328","display_name":"Van Y\u00fcz\u00fcnc\u00fc Y\u0131l University Academic Data Management System","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:a825d546b31d4cebb61fa3f97c3be97c","is_oa":true,"landing_page_url":"https://doaj.org/article/a825d546b31d4cebb61fa3f97c3be97c","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 10, Iss 2, p 58 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/10/2/58/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms10020058","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 10; Issue 2; Pages: 58","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms10020058","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10020058","pdf_url":"https://www.mdpi.com/2075-1680/10/2/58/pdf?version=1618461427","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/W3154349298.pdf","grobid_xml":"https://content.openalex.org/works/W3154349298.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W146218766","https://openalex.org/W246922367","https://openalex.org/W317665726","https://openalex.org/W649455416","https://openalex.org/W769504597","https://openalex.org/W1427293421","https://openalex.org/W1607633954","https://openalex.org/W2030959812","https://openalex.org/W2033083367","https://openalex.org/W2047073226","https://openalex.org/W2058860728","https://openalex.org/W2074412622","https://openalex.org/W2075627859","https://openalex.org/W2082048800","https://openalex.org/W2162829561","https://openalex.org/W2178624181","https://openalex.org/W2259227674","https://openalex.org/W2290568416","https://openalex.org/W2324412379","https://openalex.org/W2380376896","https://openalex.org/W2414555924","https://openalex.org/W2476507202","https://openalex.org/W2509076983","https://openalex.org/W2614636056","https://openalex.org/W2780402062","https://openalex.org/W2897178276","https://openalex.org/W2897233713","https://openalex.org/W2897299137","https://openalex.org/W2899935815","https://openalex.org/W2942439582","https://openalex.org/W3090506739","https://openalex.org/W3135485615","https://openalex.org/W3209311438","https://openalex.org/W4211202937","https://openalex.org/W4253740803","https://openalex.org/W4395109271","https://openalex.org/W6669158791","https://openalex.org/W6748396876","https://openalex.org/W6755495852","https://openalex.org/W6802979548"],"related_works":["https://openalex.org/W2082446734","https://openalex.org/W4313560640","https://openalex.org/W4246784705","https://openalex.org/W2253351617","https://openalex.org/W2065284654","https://openalex.org/W4390948881","https://openalex.org/W2164703465","https://openalex.org/W2168383058","https://openalex.org/W2143247663","https://openalex.org/W2050693590"],"abstract_inverted_index":{"The":[0,36],"purpose":[1],"of":[2,11,13,75],"this":[3],"paper":[4],"is":[5],"to":[6,80],"investigate":[7],"some":[8,54],"qualitative":[9],"properties":[10,26],"solutions":[12],"nonlinear":[14],"fractional":[15,23],"retarded":[16],"Volterra":[17],"integro-differential":[18],"equations":[19],"(FrRIDEs)":[20],"with":[21],"Caputo":[22],"derivatives.":[24],"These":[25],"include":[27],"uniform":[28],"stability,":[29,31],"asymptotic":[30],"Mittag\u2013Leffer":[32],"stability":[33],"and":[34,47,67],"boundedness.":[35],"presented":[37],"results":[38,55],"are":[39,57,62,86],"proved":[40],"by":[41],"defining":[42],"an":[43],"appropriate":[44],"Lyapunov":[45],"function":[46],"applying":[48],"the":[49,60,65,73,76,82],"Lyapunov\u2013Razumikhin":[50],"method":[51],"(LRM).":[52],"Hence,":[53],"that":[56],"available":[58],"in":[59],"literature":[61],"improved":[63],"for":[64],"FrRIDEs":[66],"obtained":[68],"under":[69],"weaker":[70],"conditions":[71],"via":[72],"advantage":[74],"LRM.":[77],"In":[78],"order":[79],"illustrate":[81],"results,":[83],"two":[84],"examples":[85],"provided.":[87]},"counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":10},{"year":2021,"cited_by_count":10}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
