Two Different Sequences of Infinitely Many Homoclinic Solutions for a Class of Fractional Hamiltonian Systems

We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems (FHS):

$$\begin{array}{c}{-}_{t}{D}_{\infty }^{\alpha }\left.{(}_{-\infty }{D}_{t}^{\alpha }x\left(t\right)\right)-L\left(t\right)x\left(t\right)+\nabla W\left(t,x\left(t\right)\right)=0,\\ x \in {H}^{\alpha }\left({\mathbb{R}},{\mathbb{R}}^{N}\right),\end{array}$$

where $\alpha \in \left(\left.\frac{1}{2},1\right]\right.,t\in {\mathbb{R}},x\in {\mathbb{R}}^{N},$ and ${-}_{t}{D}_{t}^{\alpha }$ and ${}_{t}{D}_{\infty }^{\alpha }$ are the left and right Liouville–Weyl fractional derivatives of order α on the entire axis ℝ, respectively. The novelty of our results is that, under the assumption that the nonlinearity $W\in {C}^{1}\left({\mathbb{R}}\times {\mathbb{R}}^{N},{\mathbb{R}}\right)$ involves a combination of superquadratic and subquadratic terms, for the first time, we show that the FHS possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of the FHS goes to infinity and zero, respectively. Some recent results available in the literature are generalized and significantly improved.