THREE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITION

Abstract

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^{\alpha} \left({}_0^c D_t^{\alpha}u(t)\right) +a(t)u(t)&= \lambda f(t,u(t)), \quad t\neq t_j,\ \text{a.e. } t \in [0, T], \Delta\left({}_t D_T^{\alpha-1} \left({}_0^c D_t^{\alpha}u\right)\right)(t_j)&= I_j(u(t_j)),\quad j=1,\ldots 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 applicabilityof our results.