*(English)*Zbl 1219.34009

Summary: We study the following boundary value problem of the fractional order differential equation

where $1<\alpha \le 2$, $g\in C\left(\right(0,1),[0,+\infty \left)\right)$ and $g$ may be singular at $t=0$ or/and at $t=1$, ${D}_{{0}^{+}}^{\alpha}$ is the standard Riemann-Liouville differentiation, $h\in {L}^{1}[0,1]$ is nonnegative, and $f\in C\left(\right[0,1]\times [0,+\infty ),[0,+\infty \left)\right)$.

The expression and properties of Greenâ€™s function are studied and employed to obtain some results on the existence of positive solutions by using a fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.

##### MSC:

34A08 | Fractional differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ODE |

34B18 | Positive solutions of nonlinear boundary value problems for ODE |

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

##### References:

[1] | |

[2] | |

[3] | |

[4] | |

[5] | |

[6] | |

[7] | |

[8] | |

[9] | |

[10] | |

[11] | |

[12] | |

[13] | |

[14] | |

[15] | |

[16] | |

[17] | |

[18] | |

[19] | |

[20] | |

[21] | |

[22] | |

[23] | |

[24] | |

[25] | |

[26] | |

[27] | |

[28] | |

[29] | |

[30] | |

[31] |