Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak–Sobolev spaces

In this paper, we develop some properties of the \(a_{x,y}(.)\) $a_{x,y}(.)$ -Neumann derivative for the fractional \(a_{x,y}(.)\) $a_{x,y}(.)$ -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.