In this article we are concerned in the existence and multiplicity of solutions for the following fractional Hamiltonian system 1 \(\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in {\mathbb {R}}\\ u\in H^{\alpha }({\mathbb {R}},{\mathbb {R}}^{N}), \end{array}\right. \end{aligned}\) where \(_{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are left and right Liouville–Weyl fractional derivatives of order \(\alpha \in ]\frac{1}{2},1[\) on the whole axis \({\mathbb {R}}\) respectively, \(L\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})\) is a symmetric matrix-valued function and \(W\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})\) is of subquadratic growth at infinity. Using variational methods, the minimization theorem and generalized Clark’s theorem, some results extending and improving recent results in the literature are obtained.
