MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
THREE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH DIRICHLET BOUNDARY CONDITION
G. A. Afrouzi, S. Moradi

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 ın [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 $\alpha ın (\frac{1}{2}, 1]$, $a ın C([0, T ])$ 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.

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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:  1$-$15