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**Positive solutions to boundary value problems of nonlinear fractional differential equations.**
*(English)*
Zbl 1210.34009

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 \(2<\alpha\leq 3\) 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.

\[ D^\alpha_{0+} u(t)+\lambda f(u(t))=0,\quad 0<t<1, \]

\[ u(0)=u(1)=u'(0)=0, \]

where \(2<\alpha\leq 3\) 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:

34A08 | Fractional ordinary differential equations |

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

34B09 | Boundary eigenvalue problems for ordinary differential equations |

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\textit{Y. Zhao} et al., Abstr. Appl. Anal. 2011, Article ID 390543, 16 p. (2011; Zbl 1210.34009)

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