{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:36:40Z","timestamp":1753886200381,"version":"3.41.2"},"reference-count":27,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,1,29]],"date-time":"2012-01-29T00:00:00Z","timestamp":1327795200000},"content-version":"vor","delay-in-days":28,"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":["11071141","ZR2010AM017"],"award-info":[{"award-number":["11071141","ZR2010AM017"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["11071141","ZR2010AM017"],"award-info":[{"award-number":["11071141","ZR2010AM017"]}]}],"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>By employing a well\u2010known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth\u2010order singular differential equation <jats:italic>L<\/jats:italic><jats:italic>u<\/jats:italic> = <jats:italic>p<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>f<\/jats:italic>(<jats:italic>t<\/jats:italic>, <jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic>), <jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(<jats:italic>t<\/jats:italic>)) \u2212 <jats:italic>g<\/jats:italic>(<jats:italic>t<\/jats:italic>, <jats:italic>u<\/jats:italic>(<jats:italic>t<\/jats:italic>), <jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(<jats:italic>t<\/jats:italic>)), 0 &lt; <jats:italic>t<\/jats:italic> &lt; 1, <jats:italic>\u03b1<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>u<\/jats:italic>(0) \u2212 <jats:italic>\u03b2<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>u<\/jats:italic><jats:italic>\u2032<\/jats:italic>(0) = 0, <jats:italic>\u03b3<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>u<\/jats:italic>(1) + <jats:italic>\u03b4<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>u<\/jats:italic><jats:italic>\u2032<\/jats:italic>(1) = 0, <jats:italic>\u03b1<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(0) \u2212 <jats:italic>\u03b2<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(0) = 0, <jats:italic>\u03b3<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(1) + <jats:italic>\u03b4<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>(1) = 0, with <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>\u03b4<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> \u2265 0 and <jats:italic>\u03b2<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> + <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>\u03b3<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> + <jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>\u03b4<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> &gt; 0, <jats:italic>\u2009\u2009<\/jats:italic><jats:italic>\u2009\u2009<\/jats:italic><jats:italic>i<\/jats:italic> = 1,2, where <jats:italic>L<\/jats:italic> denotes the linear operator <jats:italic>L<\/jats:italic><jats:italic>u<\/jats:italic> :  = (<jats:italic>r<\/jats:italic><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>)<jats:italic>\u2032<\/jats:italic> \u2212 <jats:italic>q<\/jats:italic><jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><jats:italic>\u2032<\/jats:italic><\/jats:sup>, <jats:italic>r<\/jats:italic> \u2208 <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup>([0,1], (0, +<jats:italic>\u221e<\/jats:italic>)), and <jats:italic>q<\/jats:italic> \u2208 <jats:italic>C<\/jats:italic>([0,1], [0, +<jats:italic>\u221e<\/jats:italic>)). This equation is viewed as a perturbation of the fourth\u2010order Sturm\u2010Liouville problem, where the perturbed term <jats:italic>g<\/jats:italic> : (0,1)\u00d7[0, +<jats:italic>\u221e<\/jats:italic>)\u00d7(\u2212<jats:italic>\u221e<\/jats:italic>, +<jats:italic>\u221e<\/jats:italic>)\u2192(\u2212<jats:italic>\u221e<\/jats:italic>, +<jats:italic>\u221e<\/jats:italic>) only satisfies the global Carath\u00e9odory conditions, which implies that the perturbed effect of <jats:italic>g<\/jats:italic> on <jats:italic>f<\/jats:italic> is quite large so that the nonlinearity can tend to negative infinity at some singular points.<\/jats:p>","DOI":"10.1155\/2012\/160891","type":"journal-article","created":{"date-parts":[[2012,1,29]],"date-time":"2012-01-29T21:00:28Z","timestamp":1327870828000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Multiple Positive Solutions of Singular Nonlinear Sturm\u2010Liouville Problems with Carath\u00e9odory Perturbed Term"],"prefix":"10.1155","volume":"2012","author":[{"given":"Yuefeng","family":"Han","sequence":"first","affiliation":[]},{"given":"Xinguang","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Lishan","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Yonghong","family":"Wu","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,1,29]]},"reference":[{"key":"e_1_2_5_1_2","doi-asserted-by":"publisher","DOI":"10.1080\/00036819508840401"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(84)90063-4"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(84)90058-X"},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.57262\/die\/1372191617"},{"key":"e_1_2_5_5_2","first-page":"305","article-title":"Positive solutions of fourth-order two-point boundary value problems","volume":"15","author":"Ma R.","year":"1999","journal-title":"Annals of Differential Equations"},{"key":"e_1_2_5_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(91)90363-5"},{"key":"e_1_2_5_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(88)90149-7"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1080\/00036818808839715"},{"key":"e_1_2_5_9_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1988-0958062-3"},{"key":"e_1_2_5_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(86)80006-3"},{"key":"e_1_2_5_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2007.08.048"},{"key":"e_1_2_5_12_2","doi-asserted-by":"publisher","DOI":"10.2307\/2048482"},{"key":"e_1_2_5_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(02)00857-3"},{"key":"e_1_2_5_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2009.04.001"},{"key":"e_1_2_5_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(03)00545-6"},{"key":"e_1_2_5_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.03.029"},{"volume-title":"Differential Equations with Boundary-Value Problems","year":"2001","author":"Zill D. 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