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Peru ́

#### Abstract

In this work we investigate the following fractional Hamiltonian systems %\begin{eqnarray}\label{eq00} $_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = \nabla W(t,u(t))$, %\end{eqnarray} where $\alpha \in (1/2, 1)$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a positive definite symmetric matrix, $W(t,u) = a(t)V(t)$ with $a\in C(\mathbb{R},\mathbb{R}^{+})$ and $V\in C^{1}(\mathbb{R}^{n}, \mathbb{R})$. By using the Mountain pass theorem and assuming that there exist $M>0$ such that $(L(t)u,u)\geq M|u|^{2}$ for all $(t,u)\in \mathbb{R}\times \mathbb{R}^{n}$ and $V$ satisfies the global Ambrosetti-Rabinowitz condition and other suitable conditions, we prove that the above mentioned equation at least has one nontrivial weak solution.

#### Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/pfda/050303

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