Comparison and sub-supersolution principles for the fractional p(x)-Laplacian

In this paper we study the new fractional Sobolev space Ws,q(x),p(x,y), where q and p are variable exponents and s∈(0,1), and the related nonlocal operator, which is a fractional version of the nonhomogeneous p(x)-Laplace operator. We first give some further qualitative properties of Ws,q(x),p(x,y). We also show the strong comparison principle for the fractional p(x)-Laplace operator. A sub-super-solution for the nonlocal equations involving the fractional p(x)-Laplacian is established.