{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:43:01Z","timestamp":1740134581526,"version":"3.37.3"},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","award":["CB-2010\/150532"],"award-info":[{"award-number":["CB-2010\/150532"]}],"id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>Given a planar system of nonautonomous ordinary differential equations,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:mi>d<\/mml:mi><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, conditions are given for the existence of an associative commutative unital algebra<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi mathvariant=\"double-struck\">A<\/mml:mi><\/mml:mrow><\/mml:math>with unit<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:math>and a function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>H<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi><mml:mo>\u2282<\/mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>\u00d7<\/mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>\u2192<\/mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math>on an open set<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi><\/mml:mrow><\/mml:math>such that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mi>e<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>w<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>and the maps<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msub><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:msub><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>are Lorch differentiable with respect to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi mathvariant=\"double-struck\">A<\/mml:mi><\/mml:mrow><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>\u03c4<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><\/mml:math>represent variables in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi mathvariant=\"double-struck\">A<\/mml:mi><\/mml:mrow><\/mml:math>. Under these conditions the solutions<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of the differential equation<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>d<\/mml:mi><mml:mi>\u03be<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mi>d<\/mml:mi><mml:mi>\u03c4<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03c4<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>over<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi mathvariant=\"double-struck\">A<\/mml:mi><\/mml:mrow><\/mml:math>define solutions<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>y<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>\u03be<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mi>e<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of the planar system.<\/jats:p>","DOI":"10.1155\/2015\/632150","type":"journal-article","created":{"date-parts":[[2015,11,9]],"date-time":"2015-11-09T16:03:33Z","timestamp":1447085013000},"page":"1-10","source":"Crossref","is-referenced-by-count":1,"title":["Algebrization of Nonautonomous Differential Equations"],"prefix":"10.1155","volume":"2015","author":[{"given":"Mar\u00eda Aracelia","family":"Alcorta-Garc\u00eda","sequence":"first","affiliation":[{"name":"Universidad Aut\u00f3noma de Nuevo Le\u00f3n, Avenida Universidad, S\/N, Ciudad Universitaria, 66451 San Nicol\u00e1s de los Garza, NL, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mart\u00edn Eduardo","family":"Fr\u00edas-Armenta","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Sonora, Boulevard Rosales y Luis Encinas, S\/N, Col. 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