  In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann
derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we
prove the basic proprieties of the correspondent function spaces. In the second
part of this paper, by means of Ekeland's variational principal and direct
variational approach, we prove the existence of weak solutions for a nonlocal
problem with nonhomogeneous Neumann and Robin boundary condition.
