Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations.

*(English)*Zbl 1186.34009The paper studies some existence results for non-local boundary value problems for fractional differential equations of order greater than unity. Following preliminaries in which definitions are provided, the main part of the paper provides two results on existence and uniqueness of solutions obtained using contraction mappings and the Krasnoselskii fixed point theorem. The paper concludes with some examples.

Reviewer: Neville Ford (Chester)

##### MSC:

34A08 | Fractional ordinary differential equations |

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

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

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\textit{B. Ahmad} and \textit{J. J. Nieto}, Abstr. Appl. Anal. 2009, Article ID 494720, 9 p. (2009; Zbl 1186.34009)

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##### References:

[1] | B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear functional differential equation with three-point boundary value problem,” preprint. · Zbl 1205.34003 |

[2] | B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009. · Zbl 1167.45003 |

[3] | B. Ahmad and V. Otero-Espinar, “Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions,” Article ID 625347, Boundary Value Problems. In press. · Zbl 1172.34004 |

[4] | B. Ahmad and S. Sivasundaram, “Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions,” Communications in Applied Analysis, vol. 13, pp. 121-228, 2009. · Zbl 1180.34003 |

[5] | J. Allison and N. Kosmatov, “Multi-point boundary value problems of fractional order,” Communications in Applied Analysis, vol. 12, pp. 451-458, 2008. · Zbl 1184.34012 |

[6] | D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3692-3705, 2008. · Zbl 1166.34033 |

[7] | B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79-88, 2007. · Zbl 1120.34323 |

[8] | Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605-609, 2009. · Zbl 1165.34313 |

[9] | V. Gafiychuk, B. Datsko, and V. Meleshko, “Mathematical modeling of time fractional reaction-diffusion systems,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 215-225, 2008. · Zbl 1152.45008 |

[10] | V. Daftardar-Gejji and S. Bhalekar, “Boundary value problems for multi-term fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 754-765, 2008. · Zbl 1151.26004 |

[11] | R. W. Ibrahim and M. Darus, “Subordination and superordination for univalent solutions for fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 871-879, 2008. · Zbl 1147.30009 |

[12] | 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 Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 |

[13] | S. Ladaci, J. J. Loiseau, and A. Charef, “Fractional order adaptive high-gain controllers for a class of linear systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 4, pp. 707-714, 2008. · Zbl 1221.93128 |

[14] | M. P. Lazarević, “Finite time stability analysis of PD\alpha fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269-279, 2006. · Zbl 1192.70008 |

[15] | I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 |

[16] | S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553-558, 2008. · Zbl 1217.81068 |

[17] | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003 |

[18] | V. A. Ilin and E. I. Moiseev, “Nonlocal boundary value problem of the first kind for a Sturm Liouville operator in its differential and finite difference aspects,” Differential Equations, vol. 23, pp. 803-810, 1987. · Zbl 0668.34025 |

[19] | V. A. Ilin and E. I. Moiseev, “Nonlocal boundary value problem of the second kind for a Sturm Liouville operator,” Differential Equations, vol. 23, pp. 979-987, 1987. · Zbl 0668.34024 |

[20] | B. Ahmad, “Approximation of solutions of the forced Duffing equation with m-point boundary conditions,” Communications in Applied Analysis, vol. 13, pp. 11-20, 2009. · Zbl 1182.34011 |

[21] | Y.-K. Chang, J. J. Nieto, and W.-S. Li, “On impulsive hyperbolic differential inclusions with nonlocal initial conditions,” Journal of Optimization Theory and Applications, vol. 140, no. 3, pp. 431-442, 2009. · Zbl 1159.49042 |

[22] | Y.-K. Chang, J. J. Nieto, and W. S. Li, “Controllability of semi-linear differential systems with nonlocal initial conditions in Banach spaces,” inpress Journal of Optimization Theory and Applications. · Zbl 1178.93029 |

[23] | P. W. Eloe and B. Ahmad, “Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol. 18, no. 5, pp. 521-527, 2005. · Zbl 1074.34022 |

[24] | J. R. Graef and J. R. L. Webb, “Third order boundary value problems with nonlocal boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1542-1551, 2009. · Zbl 1189.34034 |

[25] | P. Gurevich, “Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions,” Journal of Differential Equations, vol. 245, no. 5, pp. 1323-1355, 2008. · Zbl 1161.35008 |

[26] | R. Ma, “Positive solutions of a nonlinear m-point boundary value problem,” Computers & Mathematics with Applications, vol. 42, no. 6-7, pp. 755-765, 2001. · Zbl 0987.34018 |

[27] | R. Ma, “Multiple positive solutions for nonlinear m-point boundary value problems,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 249-262, 2004. · Zbl 1046.34030 |

[28] | V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009. · Zbl 1188.37002 |

[29] | D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980. · Zbl 0427.47036 |

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