New Results for Fractional Hamiltonian Systems

In this paper, we study the multiplicity of weak nonzero solutions for the following fractional Hamiltonian systems: \(\begin{aligned} \left\{ \begin{array}{ll} { }_{t} D_{\infty }^{\alpha }\left( _{-\infty } D_{t}^{\alpha } u(t)\right) -L(t)u +\lambda u+ \nabla W(t,u) = 0, &{} \\ u\in H^{\alpha }({\mathbb {R}},{\mathbb {R}}^N),\;\;t\in {\mathbb {R}}, &{} \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}_t{D}_{\infty }^{\alpha }\left({}_{-\infty },D_t^{\alpha },u,(t)\right)-L(t)u+\lambda u+\mathrm{\nabla }W(t,u)=0,\hfill & \hfill \\ u\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^N),t\in \mathbb{R},\hfill & \hfill \end{array}\right)\end{array}$$ where \(\alpha \in (\frac{1}{2},1],\) $\alpha \in (\frac{1}{2},1],$ \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ , \({}_{-\infty }D^{\alpha }_{t}\) ${}_{-\infty }{D}_t^{\alpha }$ and \({}_{t}D^{\alpha }_{\infty }\) ${}_t{D}_{\infty }^{\alpha }$ are left and right Liouville–Weyl fractional derivatives of order \(\alpha \) $\alpha$ on real line \({\mathbb {R}},\) $\mathbb{R},$ the matrix L(t) is not necessarily coercive nor uniformly positive definite and \(W: {\mathbb {R}}\times {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) $W:\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ satisfies some new general and weak conditions. Our results are proved using new symmetric mountain pass theorem established by Kajikia. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.