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Zbl 1210.34009
Zhao, Yige; Sun, Shurong; Han, Zhenlai; Li, Qiuping
Positive solutions to boundary value problems of nonlinear fractional differential equations.
(English)
[J] Abstr. Appl. Anal. 2011, Article ID 390543, 16 p. (2011). ISSN 1085-3375; ISSN 1687-0409/e

Summary: We study the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation $$D^\alpha_{0+} u(t)+\lambda f(u(t))=0,\quad 0<t<1,$$ $$u(0)=u(1)=u'(0)=0,$$ where $2<\alpha\le 3$ is a real number, $D^\alpha_{0+}$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter, and $f:(0,+\infty)\to (0,+\infty)$ is continuous. By the properties of the Green function and the Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are determined, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. Some examples are presented to illustrate the main results.
MSC 2000:
*34A08
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations
34B09 Boundary value problems with an indefinite weight

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