##
**Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions.**
*(English)*
Zbl 1207.34007

Summary: We study a Dirichlet boundary value problem for the Langevin equation involving two fractional orders. The Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, the ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of the Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

### MSC:

34A08 | Fractional ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

PDFBibTeX
XMLCite

\textit{B. Ahmad} and \textit{J. J. Nieto}, Int. J. Differ. Equ. 2010, Article ID 649486, 10 p. (2010; Zbl 1207.34007)

### References:

[1] | 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 B. V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 |

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

[3] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif., USA, 1999. · Zbl 0924.34008 |

[4] | 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 · doi:10.1155/2009/708576 |

[5] | B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009. · Zbl 1186.34009 · doi:10.1155/2009/494720 |

[6] | 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 · doi:10.1016/j.camwa.2009.07.091 |

[7] | B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory,” to appear in Topological Methods in Nonlinear Analysis. · Zbl 1245.34008 |

[8] | B. Ahmad, “Existence of solutions for irregular boundary value problems involving nonlinear fractional differential equations,” Applied Mathematics Letters, 2009. · Zbl 1198.34007 · doi:10.1016/j.aml.2009.11.004 |

[9] | B. Ahmad and J. J. Nieto, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional order,” to appear in Taiwanese Journal of Mathematics. · Zbl 1270.34034 |

[10] | 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 · doi:10.1016/j.mcm.2008.03.014 |

[11] | 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 · doi:10.1016/j.jmaa.2008.04.065 |

[12] | R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0998.26002 |

[13] | 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 · doi:10.1016/j.physleta.2007.06.071 |

[14] | A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis, vol. 72, pp. 580-586, 2010. · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106 |

[15] | D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Fractional Nambu mechanics,” International Journal of Theoretical Physics, vol. 48, no. 4, pp. 1044-1052, 2009. · Zbl 1170.70009 · doi:10.1007/s10773-008-9877-9 |

[16] | M. R. Ubriaco, “Entropies based on fractional calculus,” Physics Letters A, vol. 373, no. 30, pp. 2516-2519, 2009. · Zbl 1231.82024 · doi:10.1016/j.physleta.2009.05.026 |

[17] | W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, vol. 14 of World Scientific Series in Contemporary Chemical Physics, World Scientific, River Edge, NJ, USA, 2nd edition, 2004. · Zbl 1098.82001 |

[18] | S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309-6320, 2008. · Zbl 1225.82049 · doi:10.1016/j.physleta.2008.08.045 |

[19] | S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol. 42, no. 6, Article ID 065208, 34 pages, 2009. · Zbl 1156.82010 · doi:10.1088/1751-8113/42/6/065208 |

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

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.