Abstract:

Let $n,l,k$ be positive integers and $\alpha\in (n-1,n)$, $\beta\in (l-1,l)$ and $\gamma\in (k-1,k)$. Firstly the continuous general solutions of the Langevin equation with three fractional derivatives $[D_{0^+}^\alpha D_{0^+}^\beta -\lambda D_{0^+}^\gamma ] x(t)=P(t)$ are presented by using iterative method. Secondly the piecewise continuous general solutions of the impulsive Langevin equation with three fractional derivatives $[D_{0^+}^\alpha D_{0^+}^\beta -\lambda D_{0^+}^\gamma ] x(t)=P(t),t\in (t_i,t_{i+1}],i\in {\Bbb N} _0^m$ are given by using mathematical in{\rm d}uction method. Thirdly, by using the obtained results, a boundary value problem for the impulsive Langevin fractional differential equation with three Riemann-Liouville fractional derivatives of order $\alpha,\beta\in (1,2),\gamma\in (0,1)$ is converted to an integral equation. Existence results for solutions of the mentioned problem are established. Some examples are given to show readers the applications of the main results.

Key words: Iterative method, Impulsive multi-term fractional Langevin equation, Riemann-Liouville fractional derivative, Boundary value problem, Integral equation