Three solutions for impulsive fractional differential equations with Dirichlet boundary condition

G. A. Afrouzi, S. Moradi

In this paper, we discuss the existence of at least three weak solutions for the following impulsive nonlinear fractional boundary value problem \begin{align*} {}_t D_T^{lpha} eft({}_0^c D_t^{lpha}u(t)\right) +a(t)u(t)&= ambda f(t,u(t)), \quad teq t_j, ext{a.e. } t ı [0, T], \Deltaeft({}_t D_T^{lpha-1} eft({}_0^c D_t^{lpha}u\right)\right)(t_j)&= I_j(u(t_j)),\quad j=1,... n, u(0) = u(T) &= 0 \end{align*} where $\alpha \in (\frac{1}{2}, 1]$, $a \in C([0, T ])$ and $f : [0, T ]\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function. Our technical approach is based on variational methods. An example is provided to illustrate the applicability of our results.