Bifurcation results for a class of fractional Hamiltonian systems with Liouville–Weyl fractional derivatives

Abstract

In this paper, by using variational methods, we prove the existence for a class of fractional Hamiltonian systems with Liouville–Weyl fractional derivatives $\{t{D}_{\infty }^{\alpha }(-\infty D_t^{\alpha }u(t))+b(t)u(t)-\lambda u(t)=\mu f(t,u(t)),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}t\in ℝu\in H\alpha (ℝ)$ where $\alpha \in (1/2,1],t{D}_{\infty }^{\alpha }$ and $-\infty D_t^{\alpha }$ are the right and left inverse operators of the corresponding Liouville–Weyl fractional integrals of order α respectively, $u\in ℝ,b:ℝ\to ℝ,b\in L\infty (ℝ)$ , $inft\in ℝb(t)>0,\lambda ,\mu$ are real parameters and $f:ℝ\times ℝ\to ℝ$ is a function that satisfies some suitable conditions.