{"id":"https://openalex.org/W4289860774","doi":"https://doi.org/10.3390/sym14081606","title":"Stability of Nonlinear Fractional Delay Differential Equations","display_name":"Stability of Nonlinear Fractional Delay Differential Equations","publication_year":2022,"publication_date":"2022-08-04","ids":{"openalex":"https://openalex.org/W4289860774","doi":"https://doi.org/10.3390/sym14081606"},"language":"en","primary_location":{"id":"doi:10.3390/sym14081606","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081606","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1606/pdf?version=1659671352","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","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/8/1606/pdf?version=1659671352","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000832083","display_name":"D. A. Refaai","orcid":null},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"D. A. Refaai","raw_affiliation_strings":["Department of Mathematics, Collage for Women, Ain Shams University, Cairo 11566, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Collage for Women, Ain Shams University, Cairo 11566, Egypt","institution_ids":["https://openalex.org/I107720978"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067888151","display_name":"M. M. A. El-Sheikh","orcid":"https://orcid.org/0000-0002-4233-3415"},"institutions":[{"id":"https://openalex.org/I63601056","display_name":"Menoufia University","ror":"https://ror.org/05sjrb944","country_code":"EG","type":"education","lineage":["https://openalex.org/I63601056"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"M. M. A. El-Sheikh","raw_affiliation_strings":["Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt","institution_ids":["https://openalex.org/I63601056"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022135523","display_name":"Gehan A. Ismail","orcid":"https://orcid.org/0000-0002-3996-0047"},"institutions":[{"id":"https://openalex.org/I107720978","display_name":"Ain Shams University","ror":"https://ror.org/00cb9w016","country_code":"EG","type":"education","lineage":["https://openalex.org/I107720978"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Gamal A. F. Ismail","raw_affiliation_strings":["Department of Mathematics, Collage for Women, Ain Shams University, Cairo 11566, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Collage for Women, Ain Shams University, Cairo 11566, Egypt","institution_ids":["https://openalex.org/I107720978"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060547119","display_name":"Mohammed Zakarya","orcid":"https://orcid.org/0000-0003-4312-8330"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]},{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Mohammed Zakarya","raw_affiliation_strings":["Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia","Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"],"raw_orcid":"https://orcid.org/0000-0003-4312-8330","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002110465","display_name":"Ghada AlNemer","orcid":"https://orcid.org/0000-0002-2222-7973"},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Ghada AlNemer","raw_affiliation_strings":["Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-2222-7973","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004188579","display_name":"Haytham M. Rezk","orcid":"https://orcid.org/0000-0001-6782-7908"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Haytham M. Rezk","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"],"raw_orcid":"https://orcid.org/0000-0001-6782-7908","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt","institution_ids":["https://openalex.org/I184834183"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5000832083","https://openalex.org/A5002110465"],"corresponding_institution_ids":["https://openalex.org/I106778892","https://openalex.org/I107720978"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.0675,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.73483768,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"14","issue":"8","first_page":"1606","last_page":"1606"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12508","display_name":"Functional Equations Stability Results","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T12508","display_name":"Functional Equations Stability Results","score":0.9998000264167786,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9983999729156494,"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9976000189781189,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/gronwalls-inequality","display_name":"Gronwall's inequality","score":0.7368701696395874},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.7327653169631958},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7257036566734314},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6600910425186157},{"id":"https://openalex.org/keywords/delay-differential-equation","display_name":"Delay differential equation","score":0.5338396430015564},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5271791219711304},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.478633314371109},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.43951451778411865},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.38329318165779114},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.25581595301628113},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1737256944179535},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05617782473564148}],"concepts":[{"id":"https://openalex.org/C37608338","wikidata":"https://www.wikidata.org/wiki/Q510733","display_name":"Gronwall's inequality","level":3,"score":0.7368701696395874},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.7327653169631958},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7257036566734314},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6600910425186157},{"id":"https://openalex.org/C147504518","wikidata":"https://www.wikidata.org/wiki/Q2146125","display_name":"Delay differential equation","level":3,"score":0.5338396430015564},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5271791219711304},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.478633314371109},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.43951451778411865},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38329318165779114},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.25581595301628113},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1737256944179535},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05617782473564148},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"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":3,"locations":[{"id":"doi:10.3390/sym14081606","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081606","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1606/pdf?version=1659671352","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"},{"id":"pmh:oai:doaj.org/article:348611f9408543588652c1816fd47953","is_oa":true,"landing_page_url":"https://doaj.org/article/348611f9408543588652c1816fd47953","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":"Symmetry, Vol 14, Iss 8, p 1606 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/8/1606/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14081606","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14081606","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14081606","pdf_url":"https://www.mdpi.com/2073-8994/14/8/1606/pdf?version=1659671352","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":[{"score":0.5600000023841858,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4289860774.pdf","grobid_xml":"https://content.openalex.org/works/W4289860774.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W626717426","https://openalex.org/W1532178215","https://openalex.org/W1547399276","https://openalex.org/W1561829460","https://openalex.org/W1974688961","https://openalex.org/W1987702649","https://openalex.org/W1993814591","https://openalex.org/W1997441517","https://openalex.org/W2002669440","https://openalex.org/W2004169635","https://openalex.org/W2021492437","https://openalex.org/W2038435020","https://openalex.org/W2078055801","https://openalex.org/W2082256594","https://openalex.org/W2187289308","https://openalex.org/W2587563718","https://openalex.org/W2607192770","https://openalex.org/W2716418819","https://openalex.org/W2789959572","https://openalex.org/W2790911812","https://openalex.org/W2982014627","https://openalex.org/W3011005186","https://openalex.org/W3197087296","https://openalex.org/W3214108027","https://openalex.org/W4220759147","https://openalex.org/W4223930072","https://openalex.org/W6633805806","https://openalex.org/W6649373576","https://openalex.org/W6669993287","https://openalex.org/W6733653382","https://openalex.org/W6736751745","https://openalex.org/W7006382482"],"related_works":["https://openalex.org/W1261320338","https://openalex.org/W2056281781","https://openalex.org/W2942981489","https://openalex.org/W2924035449","https://openalex.org/W2086307616","https://openalex.org/W4243954292","https://openalex.org/W2903509708","https://openalex.org/W2977631248","https://openalex.org/W2753150188","https://openalex.org/W2389579716"],"abstract_inverted_index":{"This":[0],"article":[1],"discusses":[2],"several":[3],"forms":[4],"of":[5,8],"Ulam":[6],"stability":[7,33],"nonlinear":[9],"fractional":[10],"delay":[11],"differential":[12],"equations.":[13],"Our":[14],"investigation":[15],"is":[16],"based":[17],"on":[18],"a":[19],"generalised":[20],"Gronwall\u2019s":[21],"inequality":[22],"and":[23],"Picard":[24],"operator":[25],"theory.":[26],"Implementations":[27],"are":[28],"provided":[29],"to":[30],"demonstrate":[31],"the":[32],"results":[34],"obtained":[35],"for":[36],"finite":[37],"intervals.":[38]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
