Multiplicity of solutions for fractional Hamiltonian systems with combined nonlinearities and without coercive conditions

Consider the following fractional Hamiltonian system: \(\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in \mathbb {R}\\ u\in H^{\alpha }(\mathbb {R}). \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}_t^{}{D}_{\infty }^{\alpha }{(}_{-\infty }{D}_t^{\alpha }u)(t)+L(t)u(t)=\mathrm{\nabla }W(t,u(t)),t\in \mathbb{R}\hfill \\ u\in H^{\alpha }(\mathbb{R}).\hfill \end{array}\right)\end{array}$$ Here, \(_{t}D_{\infty }^{\alpha }\) $_t^{}{D}_{\infty }^{\alpha }$ and \(_{-\infty }D_{t}^{\alpha }\) $_{-\infty }^{}{D}_t^{\alpha }$ represent the Liouville-Weyl fractional derivatives of order \(\frac{1}{2}< \alpha < 1\) $\frac{1}{2}<\alpha <1$ , \(L \in C(\mathbb {R}, \mathbb {R}^{N^2})\) $L\in C(\mathbb{R},{\mathbb{R}}^{N^2})$ is a symmetric matrix, and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^N, \mathbb {R})\) $W\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ . By applying the Fountain Theorem and the Dual Fountain Theorem, we demonstrate that this system admits two distinct sequences of solutions under the condition that L meets a new non-coercive criterion, and the potential W(t, x) exhibits combined nonlinearities.