  In this paper, we are concerned with some qualitative properties of the new
fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the
generalized Poincar\'e type inequality and some continuous and compact
embedding theorems of these spaces. Moreover, we prove that any function in
$W^sL_{\varPhi_{x,y}}(\Omega)$ may be extended to a function in
$W^sL_{\varPhi_{x,y}}(\R^N)$, with $\Omega \subset \R^N$ is a bounded domain of
class $C^{0,1}$. In addition, we establish a result relates to the complemented
subspace in $W^s{L_{\varPhi_{x,y}}}\left( \R^N\right)$. As an application,
using the mountain pass theorem and some variational methods, we investigate
the existence of a nontrivial weak solution for a class of nonlocal fractional
type problems with Dirichlet boundary data.
