  In this paper, we study the following nonlocal problem in fractional Orlicz
Sobolev spaces
  \begin{eqnarray*}
  (-\Delta_{\Phi})^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in\mathbb{R}^N,
  \end{eqnarray*}
  where $(-\Delta_{\Phi})^{s}(s\in(0, 1))$ denotes the non-local and maybe
non-homogeneous operator, the so-called fractional $\Phi$-Laplacian. Without
assuming the Ambrosetti-Rabinowitz type and the Nehari type conditions on the
nonlinearity, we obtain the existence of ground state solutions for the above
problem in periodic case. The proof is based on a variant version of the
mountain pass theorem and a Lions' type result for fractional Orlicz Sobolev
spaces.
