  In this paper, we extend the fractional Sobolev spaces with variable
exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$,
where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We
are concerned with some qualitative properties of the space $W^{K,p(x,y)}$
(completeness, reflexivity, separability, and density). Moreover, we prove a
continuous and compact embedding theorem of these spaces into variable exponent
Lebesgue spaces. As applications, we discuss the existence of a nontrivial
solution for a nonlocal $p(x,.)$-Kirchhoff type problem. Further, we establish
the existence and uniqueness of a solution for a variational problem involving
the integro-differential operator of elliptic type $\mathcal{L}^{p(x,.)}_K$.
