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 applicability of our results.

Keywords: Three solutions; fractional differential equation; impulsive effect; variational methods; critical point theory.

MSC: 26A33, 34B15, 35A15, 34B15, 34K45, 58E05

DOI: 10.57016/MV-xgfv9794

Pages:  189$-$203     

Volume  75 ,  Issue  3 ,  2023