In this study, we explore the existence of multiple small solutions for fractional differential equations of Hamiltonian type characterized by: \(\begin{aligned} \left\{ \begin{array}{ll} _{t} D_{\infty }^{\alpha }\left( _{-\infty } D_{t}^{\alpha } u(t)\right) -L(t)u + \nabla W(t,u) = 0, & \\ u\in H^{\alpha }({\mathbb {R}},{\mathbb {R}}^N),\;\;t\in {\mathbb {R}}, & \end{array} \right. \end{aligned}\) where \(\alpha \in (\frac{1}{2}, 1]\) , \({EMPTY}{-\infty }D{t}^{\alpha }\) and \({EMPTY}{t}D{\infty }^{\alpha }\) denote the left and right Liouville-Weyl fractional derivatives of order \(\alpha \) on the real line \({\mathbb {R}}\) , respectively. Using a new symmetric mountain pass theorem established by Kajikia, we prove the existence of infinitely many solutions for this system, even when the matrix L(t) is not necessarily coercive or uniformly positive definite and W(t, x) is defined only locally near the coordinate origin \(x = 0\) . The theorems proved significantly generalize and improve upon previously obtained results. We also provide several illustrative examples.
