{"id":"https://openalex.org/W4294958837","doi":"https://doi.org/10.3390/sym14091823","title":"Some Generalizations of (\u2207\u2207)\u0394\u2013Gronwall\u2013Bellman\u2013Pachpatte Dynamic Inequalities on Time Scales with Application","display_name":"Some Generalizations of (\u2207\u2207)\u0394\u2013Gronwall\u2013Bellman\u2013Pachpatte Dynamic Inequalities on Time Scales with Application","publication_year":2022,"publication_date":"2022-09-02","ids":{"openalex":"https://openalex.org/W4294958837","doi":"https://doi.org/10.3390/sym14091823"},"language":"en","primary_location":{"id":"doi:10.3390/sym14091823","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091823","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1823/pdf?version=1662107026","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/2073-8994/14/9/1823/pdf?version=1662107026","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000462219","display_name":"Ahmed A. El\u2010Deeb","orcid":null},"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":true,"raw_author_name":"Ahmed A. El-Deeb","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"],"raw_orcid":"https://orcid.org/0000-0003-2822-4092","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108729718","display_name":"Alaa A. El\u2010Bary","orcid":null},"institutions":[{"id":"https://openalex.org/I4210118384","display_name":"Academy of Scientific Research and Technology","ror":"https://ror.org/02k284p70","country_code":"EG","type":"nonprofit","lineage":["https://openalex.org/I4210118384"]},{"id":"https://openalex.org/I59272784","display_name":"Arab Academy for Science, Technology, and Maritime Transport","ror":"https://ror.org/0004vyj87","country_code":"EG","type":"education","lineage":["https://openalex.org/I59272784"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"Alaa A. El-Bary","raw_affiliation_strings":["Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria 21532, Egypt","Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, Egypt","National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria 21532, Egypt","institution_ids":["https://openalex.org/I59272784"]},{"raw_affiliation_string":"Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, Egypt","institution_ids":["https://openalex.org/I4210118384"]},{"raw_affiliation_string":"National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt","institution_ids":["https://openalex.org/I4210118384"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025074757","display_name":"Jan Awrejcewicz","orcid":"https://orcid.org/0000-0003-0387-921X"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Jan Awrejcewicz","raw_affiliation_strings":["Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland"],"raw_orcid":"https://orcid.org/0000-0003-0387-921X","affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5000462219","https://openalex.org/A5025074757"],"corresponding_institution_ids":["https://openalex.org/I184834183","https://openalex.org/I188884621"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3558,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.50152743,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"14","issue":"9","first_page":"1823","last_page":"1823"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9994999766349792,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9994999766349792,"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/T12809","display_name":"Contact Mechanics and Variational Inequalities","score":0.9868999719619751,"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"}},{"id":"https://openalex.org/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9860000014305115,"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/uniqueness","display_name":"Uniqueness","score":0.7654902935028076},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7170472741127014},{"id":"https://openalex.org/keywords/dynamic-equation","display_name":"Dynamic equation","score":0.7022280097007751},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.6040588617324829},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5934781432151794},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5668532252311707},{"id":"https://openalex.org/keywords/gronwalls-inequality","display_name":"Gronwall's inequality","score":0.5638346672058105},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.51898592710495},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.46877139806747437},{"id":"https://openalex.org/keywords/oscillation","display_name":"Oscillation (cell signaling)","score":0.4578936696052551},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.288964182138443},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23173308372497559},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1613718569278717}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7654902935028076},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7170472741127014},{"id":"https://openalex.org/C2989370511","wikidata":"https://www.wikidata.org/wiki/Q18381590","display_name":"Dynamic equation","level":3,"score":0.7022280097007751},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.6040588617324829},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5934781432151794},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5668532252311707},{"id":"https://openalex.org/C37608338","wikidata":"https://www.wikidata.org/wiki/Q510733","display_name":"Gronwall's inequality","level":3,"score":0.5638346672058105},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.51898592710495},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.46877139806747437},{"id":"https://openalex.org/C2778439541","wikidata":"https://www.wikidata.org/wiki/Q7106412","display_name":"Oscillation (cell signaling)","level":2,"score":0.4578936696052551},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.288964182138443},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23173308372497559},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1613718569278717},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14091823","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091823","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1823/pdf?version=1662107026","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:5c99847cfdce4e8d9be3aff5fc312c49","is_oa":true,"landing_page_url":"https://doaj.org/article/5c99847cfdce4e8d9be3aff5fc312c49","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 9, p 1823 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/9/1823/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14091823","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/sym14091823","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091823","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1823/pdf?version=1662107026","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":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.6800000071525574}],"awards":[],"funders":[{"id":"https://openalex.org/F4320334763","display_name":"Leibniz-Gemeinschaft","ror":"https://ror.org/01n6r0e97"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4294958837.pdf","grobid_xml":"https://content.openalex.org/works/W4294958837.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W98230489","https://openalex.org/W199863790","https://openalex.org/W271810198","https://openalex.org/W1522419168","https://openalex.org/W1966973071","https://openalex.org/W2026606194","https://openalex.org/W2063769985","https://openalex.org/W2077750044","https://openalex.org/W2080986297","https://openalex.org/W2102391226","https://openalex.org/W2109628201","https://openalex.org/W2112283700","https://openalex.org/W2115258461","https://openalex.org/W2133572184","https://openalex.org/W2598670005","https://openalex.org/W2800962038","https://openalex.org/W2943113586","https://openalex.org/W2967156670","https://openalex.org/W3119780542","https://openalex.org/W3165765870","https://openalex.org/W6641822753","https://openalex.org/W6657006391","https://openalex.org/W6677064763"],"related_works":["https://openalex.org/W2057030324","https://openalex.org/W1968731853","https://openalex.org/W2137980878","https://openalex.org/W2151813854","https://openalex.org/W2018954023","https://openalex.org/W2142669796","https://openalex.org/W2900582606","https://openalex.org/W3148257963","https://openalex.org/W2740452798","https://openalex.org/W2394006932"],"abstract_inverted_index":{"As":[0],"a":[1],"new":[2,14],"usage":[3],"of":[4,16,42,72],"Leibniz":[5],"integral":[6,33],"rule":[7],"on":[8,20,76],"time":[9,21,77],"scales,":[10],"we":[11],"proved":[12,48],"some":[13,26,70],"extensions":[15],"dynamic":[17,74,91],"Gronwall\u2013Pachpatte-type":[18],"inequalities":[19,36,47],"scales.":[22,78],"Our":[23],"results":[24,28],"extend":[25],"existing":[27],"in":[29,53,84],"the":[30,43,54,61,86],"literature.":[31],"Some":[32],"and":[34,66],"discrete":[35],"are":[37],"obtained":[38],"as":[39,56],"special":[40],"cases":[41],"main":[44],"results.":[45],"The":[46],"here":[49],"can":[50],"be":[51],"used":[52],"analysis":[55],"handy":[57],"tools":[58],"to":[59,89],"study":[60],"stability,":[62],"boundedness,":[63],"existence,":[64],"uniqueness":[65],"oscillation":[67],"behavior":[68],"for":[69],"kinds":[71],"partial":[73],"equations":[75],"Symmetry":[79],"plays":[80],"an":[81],"essential":[82],"role":[83],"determining":[85],"correct":[87],"methods":[88],"solve":[90],"inequalities.":[92]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2022-09-08T00:00:00"}
