Eur. Phys. J. Special Topics 222, 1857-1874 (2013)
https://doi.org/10.1140/epjst/e2013-01969-9

Regular Article

Presentation of solutions of impulsive fractional Langevin equations and existence results

Impulsive fractional Langevin equations

¹ School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China
² Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
³ Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia
⁴ Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
⁵ Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China

^a e-mail: sci.jrwang@gzu.edu.cn
^b e-mail: Michal.Feckan@fmph.uniba.sk
^c e-mail: yzhou@xtu.edu.cn

Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013

Abstract

In this paper, a class of impulsive fractional Langevin equations is firstly offered. Formula of solutions involving Mittag-Leffer functions and impulsive terms of such equations are successively derived by studying the corresponding linear Langevin equations with two different fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity and nonnegative of Mittag-Leffer functions and fixed point methods. Further, other existence results of nonlinear impulsive problems are also presented. Finally, an example is given to illustrate our theoretical results.