In this paper, we study the fractional p(⋅, ⋅)-Laplacian, and we introduce the corresponding nonlocal conormal derivative for this operator. We prove the basic properties of the corresponding function space, and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, see Sec. IV, we prove the existence of weak solutions of corresponding p(⋅, ⋅)-Robin boundary problems with sign-changing potentials by applying variational tools.

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