Positive solutions for multipoint boundary value problem of fractional differential equations.

*(English)*Zbl 1223.34008The author studies the existence of positive solutions for the boundary value problem

\[ \begin{cases} D^{\alpha}u(t)~+~f(t,u(t))~=~0,~~t\in (0,1) \text{~~and~~} \alpha \in(2,3),\\ u(0)~=~u^{\prime}(0)~=~0, ~~~ u^{\prime}(1)~=~\sum_{i=1}^{m-2}~a_i~u^{\prime}(\xi_i).\end{cases}\tag{*} \]

He proves in Theorem 3.4 that the problem (*) has at least one positive solution. In Theorem 3.5 he shows, under some additional conditions, that the problem (*) has at least three positive solutions.

\[ \begin{cases} D^{\alpha}u(t)~+~f(t,u(t))~=~0,~~t\in (0,1) \text{~~and~~} \alpha \in(2,3),\\ u(0)~=~u^{\prime}(0)~=~0, ~~~ u^{\prime}(1)~=~\sum_{i=1}^{m-2}~a_i~u^{\prime}(\xi_i).\end{cases}\tag{*} \]

He proves in Theorem 3.4 that the problem (*) has at least one positive solution. In Theorem 3.5 he shows, under some additional conditions, that the problem (*) has at least three positive solutions.

Reviewer: Ahmed M. A. El-Sayed (Alexandria)

##### MSC:

34A08 | Fractional ordinary differential equations |

26A33 | Fractional derivatives and integrals |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

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

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