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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\textit{W. Zhong}, Abstr. Appl. Anal. 2010, Article ID 601492, 15 p. (2010; Zbl 1223.34008)

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##### References:

[1] | K. B. Oldham and J. Spanier, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011 |

[2] | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral And Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0924.44003 |

[3] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1201.44007 |

[4] | O. P. Sabatier, J. A. Agrawal, and T. Machado, Advances in Frcational Calculus, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1196.35069 |

[5] | I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 |

[6] | Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048 |

[7] | S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1-12, 2006. · Zbl 1096.34016 |

[8] | B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838-1843, 2009. · Zbl 1205.34003 |

[9] | M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391-2396, 2009. · Zbl 1198.26007 |

[10] | C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363-1375, 2010. · Zbl 1189.34014 |

[11] | W. Zhong and W. Lin, “Nonlocal and multiple-point boundary value problem for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1345-1351, 2010. · Zbl 1189.34036 |

[12] | Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916-924, 2010. · Zbl 1187.34026 |

[13] | H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565-572, 2009. · Zbl 1176.34070 |

[14] | M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. · Zbl 1149.26012 |

[15] | C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050-1055, 2010. · Zbl 1204.34007 |

[16] | S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300-1309, 2010. · Zbl 1189.34050 |

[17] | J. R. L. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems: a unified approach,” Journal of the London Mathematical Society, vol. 74, no. 3, pp. 673-693, 2006. · Zbl 1115.34028 |

[18] | M. A. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964. · Zbl 0121.10604 |

[19] | R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673-688, 1979. · Zbl 0421.47033 |

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