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

In this paper we study the new fractional Sobolev space W-s,W-q(x),W-P(x,W-y), where q and p are variable exponents and s is an element of (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 W-s,W-q(x),W-P(x,W-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. (C) 2017 Elsevier Inc. All rights reserved.