A new fractional function space X^{k(⋅),α(⋅)}(Ω) with variable exponents k, α and its relaxed properties are established in this paper. Under this configuration, the comparison principle of the fractional α(⋅)-Laplace operator and a priori estimate of weak solutions for a variable-order fractional α(⋅)-Laplacian equation are obtained. Then, the application of a fixed point theorem ensures the existence of weak solutions for a nonlinear elliptic equation with the variable-order fractional α(⋅)-Laplacian.