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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 390543, 16 pages
Research Article

Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2Department of Mathematics and Statistics, Missouri University of Science and Technology Rolla, MO 65409-0020, USA

Received 23 September 2010; Revised 5 November 2010; Accepted 6 December 2010

Academic Editor: Josef Diblรญk

Copyright ยฉ 2011 Yige Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations ๐ท๐›ผ0+๐‘ข(๐‘ก)+๐œ†๐‘“(๐‘ข(๐‘ก))=0, 0<๐‘ก<1, ๐‘ข(0)=๐‘ข(1)=๐‘ขโ€ฒ(0)=0, where 2<๐›ผโ‰ค3 is a real number, ๐ท๐›ผ0+ is the Riemann-Liouville fractional derivative, ๐œ† is a positive parameter, and ๐‘“โˆถ(0,+โˆž)โ†’(0,+โˆž) is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.