{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:14:19Z","timestamp":1753881259838,"version":"3.41.2"},"reference-count":22,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,8,13]],"date-time":"2012-08-13T00:00:00Z","timestamp":1344816000000},"content-version":"vor","delay-in-days":225,"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":["11161018","10452408801004217"],"award-info":[{"award-number":["11161018","10452408801004217"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province","doi-asserted-by":"crossref","award":["11161018","10452408801004217"],"award-info":[{"award-number":["11161018","10452408801004217"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We study the following second\u2010order super\u2010half\u2010linear impulsive differential equations with delay [<jats:italic>r<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>\u03c6<\/jats:italic><jats:sub><jats:italic>\u03b3<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic><jats:italic>\u2032<\/jats:italic>(<jats:italic>t<\/jats:italic>))]<jats:italic>\u2032<\/jats:italic> + <jats:italic>p<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>\u03c6<\/jats:italic><jats:sub><jats:italic>\u03b3<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>(<jats:italic>t<\/jats:italic> \u2212 <jats:italic>\u03c3<\/jats:italic>)) + <jats:italic>q<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>(<jats:italic>t<\/jats:italic> \u2212 <jats:italic>\u03c3<\/jats:italic>)) = <jats:italic>e<\/jats:italic>(<jats:italic>t<\/jats:italic>), <jats:italic>t<\/jats:italic> \u2260 <jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, <jats:italic>x<\/jats:italic>(<jats:italic>t<\/jats:italic><jats:sup>+<\/jats:sup>) = <jats:italic>a<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub><jats:italic>x<\/jats:italic>(<jats:italic>t<\/jats:italic>), <jats:italic>x<\/jats:italic><jats:italic>\u2032<\/jats:italic>(<jats:italic>t<\/jats:italic><jats:sup>+<\/jats:sup>) = <jats:italic>b<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub><jats:italic>x<\/jats:italic><jats:italic>\u2032<\/jats:italic>(<jats:italic>t<\/jats:italic>),\n\t\t\t\t\t<jats:italic>t<\/jats:italic> = <jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, where <jats:italic>t<\/jats:italic> \u2265 <jats:italic>t<\/jats:italic><jats:sub>0<\/jats:sub> \u2208 <jats:italic>\u211d<\/jats:italic>, <jats:italic>\u03c6<\/jats:italic><jats:sub>*<\/jats:sub>(<jats:italic>u<\/jats:italic>) = |<jats:italic>u<\/jats:italic>|<jats:sup>*\u22121<\/jats:sup><jats:italic>u<\/jats:italic>, <jats:italic>\u03c3<\/jats:italic> is a nonnegative constant, {<jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>} denotes the impulsive moments sequence with <jats:italic>\u03c4<\/jats:italic><jats:sub>1<\/jats:sub> &lt; <jats:italic>\u03c4<\/jats:italic><jats:sub>2<\/jats:sub> &lt; \u22ef&lt;<jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> &lt; \u22ef, lim\u2009<jats:sub><jats:italic>k<\/jats:italic>\u2192<jats:italic>\u221e<\/jats:italic><\/jats:sub><jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> = <jats:italic>\u221e<\/jats:italic>, and <jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>+1<\/jats:sub> \u2212 <jats:italic>\u03c4<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> &gt; <jats:italic>\u03c3<\/jats:italic>. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.<\/jats:p>","DOI":"10.1155\/2012\/285051","type":"journal-article","created":{"date-parts":[[2012,8,13]],"date-time":"2012-08-13T21:21:14Z","timestamp":1344892874000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Interval Oscillation Criteria for Super\u2010Half\u2010Linear Impulsive Differential Equations with Delay"],"prefix":"10.1155","volume":"2012","author":[{"given":"Zhonghai","family":"Guo","sequence":"first","affiliation":[]},{"given":"Xiaoliang","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Wu-Sheng","family":"Wang","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,8,13]]},"reference":[{"key":"e_1_2_5_1_2","doi-asserted-by":"crossref","DOI":"10.1142\/0906","volume-title":"Theory of Impulsive Differential Equations","author":"Lakshimikantham V.","year":"1989"},{"volume-title":"Impulsive Differential Equations: Periodic Solution and Applications,","year":"1993","author":"Bainov D. D.","key":"e_1_2_5_2_2"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(89)90232-1"},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1997.5211"},{"key":"e_1_2_5_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-9659(96)00105-X"},{"key":"e_1_2_5_6_2","first-page":"201","article-title":"A nonoscillation result for delay differential equations with impulses","volume":"10","author":"Shen J. H.","year":"1994","journal-title":"Annals of Differential Equations"},{"key":"e_1_2_5_7_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1999.6297"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(00)00055-9"},{"key":"e_1_2_5_9_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2000.7218"},{"key":"e_1_2_5_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2007.03.003"},{"key":"e_1_2_5_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2008.10.020"},{"key":"e_1_2_5_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2009.04.072"},{"key":"e_1_2_5_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2006.08.040"},{"key":"e_1_2_5_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.12.041"},{"key":"e_1_2_5_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.02.073"},{"key":"e_1_2_5_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.10.098"},{"key":"e_1_2_5_17_2","doi-asserted-by":"publisher","DOI":"10.2307\/2160125"},{"key":"e_1_2_5_18_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1998.6259"},{"key":"e_1_2_5_19_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01324723"},{"key":"e_1_2_5_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2009.09.010"},{"volume-title":"Inequalities","year":"1964","author":"Hardy G. H.","key":"e_1_2_5_21_2"},{"key":"e_1_2_5_22_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1998.6159"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/285051.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/285051.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/285051","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T07:14:13Z","timestamp":1718090053000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/285051"}},"subtitle":[],"editor":[{"given":"Alfredo","family":"Bellen","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/285051"],"URL":"https:\/\/doi.org\/10.1155\/2012\/285051","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-04-25","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-07-02","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-08-13","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"285051"}}