{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:52Z","timestamp":1740134572253,"version":"3.37.3"},"reference-count":33,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11201008"],"award-info":[{"award-number":["11201008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>We investigate the solvability of a fully fourth-order periodic boundary value problem of the form<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>4<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msup><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>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>,<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>,<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,\u2009\u2009<\/mml:mo><\/mml:math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>i<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>i<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,\u2009\u2009<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0,1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>2,3<\/mml:mn><mml:mo>,<\/mml:mo><\/mml:math>where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>f<\/mml:mi><mml:mo>:<\/mml:mo><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><mml:mo>\u00d7<\/mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>\u2192<\/mml:mo><mml:mi mathvariant=\"double-struck\">R<\/mml:mi><\/mml:math>satisfies Carath\u00e9odory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. 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