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In this paper, we extend the fractional Sobolev spaces with variableexponents $W^{s,p(x,y)}$ 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. Weare concerned with some qualitative properties of the space $W^{K,p(x,y)}$(completeness, reflexivity, separability, and density). Moreover, we prove acontinuous and compact embedding theorem of these spaces into variable exponentLebesgue spaces. As applications, we discuss the existence of a nontrivialsolution for a nonlocal $p(x,.)$-Kirchhoff type problem. Further, we establishthe existence and uniqueness of a solution for a variational problem involvingthe integro-differential operator of elliptic type $\mathcal{L}^{p(x,.)}_K$.

General fractional Sobolev Space with variable exponent and applications to nonlocal problems