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Province","doi-asserted-by":"publisher","award":["13C084"],"award-info":[{"award-number":["13C084"]}],"id":[{"id":"10.13039\/501100005225","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>By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse:<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>x<\/mml:mi><mml:mi mathvariant=\"normal\">'<\/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>x<\/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:mi>a<\/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>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 stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c4<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/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 stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c4<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/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 stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mi mathvariant=\"normal\">'<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03b3<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/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 stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mi mathvariant=\"normal\">'<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03b3<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>,<\/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 stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">]<\/mml:mo><mml:mo>,\u2009\u2009<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>\u2260<\/mml:mo><mml:msub><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,\u2009\u2009<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>Z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>+<\/mml:mo><\/mml:mrow><\/mml:msub><mml:mo>;\u2009\u2009<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>+<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03b8<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>,\u2009\u2009<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msub><mml:mrow><mml:mi>Z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>+<\/mml:mo><\/mml:mrow><\/mml:msub><\/mml:math>. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.<\/jats:p>","DOI":"10.1155\/2014\/592513","type":"journal-article","created":{"date-parts":[[2014,3,3]],"date-time":"2014-03-03T02:58:53Z","timestamp":1393815533000},"page":"1-12","source":"Crossref","is-referenced-by-count":1,"title":["Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3500-7152","authenticated-orcid":true,"given":"Zhenguo","family":"Luo","sequence":"first","affiliation":[{"name":"Department of Mathematics, National University of Defense Technology, Changsha 410073, China"},{"name":"Department of Mathematics, Hengyang Normal University, Hengyang 421008, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liping","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hengyang Normal University, Hengyang 421008, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunhui","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hengyang Normal University, Hengyang 421008, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/741043"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2011.10.082"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2010.03.003"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2011.11.008"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2013.07.011"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2009.09.001"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2012.10.001"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2007.10.033"},{"key":"18","year":"1993"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1142\/9789812798664"},{"key":"20","year":"1997"},{"key":"9","year":"1993"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2000.7340"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(02)00288-6"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(03)00049-0"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.04.007"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2000.7248"},{"key":"15","year":"1977"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(79)90208-7"},{"issue":"4","key":"17","first-page":"294","volume":"13","year":"1984","journal-title":"Advances in Mathematics"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/592513.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/592513.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/592513.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,28]],"date-time":"2016-07-28T22:33:50Z","timestamp":1469745230000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/592513\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":20,"alternative-id":["592513","592513"],"URL":"https:\/\/doi.org\/10.1155\/2014\/592513","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}