{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:27:53Z","timestamp":1760783273282,"version":"3.41.2"},"reference-count":17,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2010,6,3]],"date-time":"2010-06-03T00:00:00Z","timestamp":1275523200000},"content-version":"vor","delay-in-days":153,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004761","name":"Natural Science Foundation of Hunan Provincial","doi-asserted-by":"crossref","award":["09JJ3005","2009JT3042"],"award-info":[{"award-number":["09JJ3005","2009JT3042"]}],"id":[{"id":"10.13039\/501100004761","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004761","name":"Natural Science Foundation of Hunan Provincial","doi-asserted-by":"crossref","award":["09JJ3005","2009JT3042"],"award-info":[{"award-number":["09JJ3005","2009JT3042"]}],"id":[{"id":"10.13039\/501100004761","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100005225","name":"Key Discipline in Hunan Province","doi-asserted-by":"crossref","award":["09JJ3005","2009JT3042"],"award-info":[{"award-number":["09JJ3005","2009JT3042"]}],"id":[{"id":"10.13039\/501100005225","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012269","name":"Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province","doi-asserted-by":"crossref","award":["09JJ3005","2009JT3042"],"award-info":[{"award-number":["09JJ3005","2009JT3042"]}],"id":[{"id":"10.13039\/501100012269","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation with  <jats:italic>p<\/jats:italic>\u2010Laplacian operator , 0 &lt; <jats:italic>t<\/jats:italic> &lt; 1, <jats:italic>u<\/jats:italic>(0) = <jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032<\/jats:italic><\/jats:sup>(1) = 0, <jats:italic>u<\/jats:italic><jats:sup><jats:italic>\u2032\u2032<\/jats:italic><\/jats:sup>(0) = 0, , where 0 &lt; <jats:italic>\u03b3<\/jats:italic> &lt; 1, 2 &lt; <jats:italic>\u03b1<\/jats:italic> &lt; 3, 0 &lt; <jats:italic>\u03c1<\/jats:italic> \u2a7d 1,  denotes the Caputo derivative, and <jats:italic>f<\/jats:italic> : [0,1]\u00d7[0, +<jats:italic>\u221e<\/jats:italic>) \u00d7 <jats:italic>R<\/jats:italic> \u2192 [0, +<jats:italic>\u221e<\/jats:italic>) is continuous function, <jats:italic>\u03d5<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>(<jats:italic>s<\/jats:italic>) = |<jats:italic>s<\/jats:italic>|<jats:sup><jats:italic>p<\/jats:italic>\u22122<\/jats:sup><jats:italic>s<\/jats:italic>, <jats:italic>p<\/jats:italic> &gt; 1,\u2009\u2009(<jats:italic>\u03d5<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>)\u2009<jats:sup>\u22121<\/jats:sup> = <jats:italic>\u03d5<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub>, 1\/<jats:italic>p<\/jats:italic> + 1\/<jats:italic>q<\/jats:italic> = 1. By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.<\/jats:p>","DOI":"10.1155\/2010\/495138","type":"journal-article","created":{"date-parts":[[2010,6,3]],"date-time":"2010-06-03T14:34:05Z","timestamp":1275575645000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with <i>p<\/i>\u2010Laplacian Operator"],"prefix":"10.1155","volume":"2010","author":[{"given":"Jinhua","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongjun","family":"Xiang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ZhiGang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,6,3]]},"reference":[{"key":"e_1_2_5_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2006.03.009"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2005.07.084"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2007.03.015"},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2006.01.004"},{"key":"e_1_2_5_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2004.03.002"},{"key":"e_1_2_5_6_2","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1996.0456"},{"key":"e_1_2_5_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.01.007"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/186928"},{"key":"e_1_2_5_9_2","first-page":"1","article-title":"Triple positive solutions for boundary value problem of a nonlinear fractional differential equation","volume":"33","author":"Dehghani R.","year":"2007","journal-title":"Bulletin of the Iranian Mathematical Society"},{"key":"e_1_2_5_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0252\u20109602(06)60044\u20101"},{"key":"e_1_2_5_11_2","first-page":"1","article-title":"Positive solutions for boundary value problems of nonlinear fractional differential equations","volume":"36","author":"Zhang S. 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