  In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev
spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)$. Next, by means of Ekeland's
variational principal, we show that there exists $\lambda_*>0$ such that any
$\lambda\in(0, \lambda_*)$ is an eigenvalue for the following problem
$$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left(
-\Delta\right)^{s(x,.)}_{a_{(x,.)}} u = \lambda |u|^{q(x)-2}u &\quad {\rm in}\
\Omega, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus \Omega,
\end{array} \right. $$ where $\Omega$ is a bounded open subset of
$\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary.
