Existence and Multiplicity of Solutions for Impulsive Fractional Differential Equations

Abstract

In this paper, we study the existence of solutions for the following impulsive fractional boundary-value problem: $$\begin{aligned} {\left\{ \begin{array}{ll} - \frac{\mathrm{d}}{\mathrm{d}t} \Big (\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t)) \Big ) = \lambda u (t) + f (t, u (t)), &{} t \ne t_j, \;\;\text {a.e.}\;\; t \in [0, T],\\ \Delta \Big (\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j)) \Big ) = I_j (u (t_j)), &{} j = 1, 2, \ldots , n,\\ u (0) = u (T) = 0, \end{array}\right. } \end{aligned}$$ where \(\alpha \in (1/2, 1]\), \(0 = t_0< t_1< t_2< \cdots< t_n< t_{n +1} = T\), \(\lambda \) is a parameter and \(f :[0, T] \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) and \(I_j : {\mathbb {R}} \rightarrow {\mathbb {R}}\), \(j = 1, \ldots , n\) are continuous functions and $$\begin{aligned}&\Delta \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j)) \right) \\&\quad = \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^+) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^+) \\&\qquad -\, \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^-) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^-) ,\\&\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^+) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^+)) \nonumber \\&\quad = \lim _{t \rightarrow t_j^+} \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t))\right) ,\\&\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^-) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^-)) \\&\quad = \lim _{t \rightarrow t_j^-} \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t))\right) . \end{aligned}$$ By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution and infinitely many solutions.