  In this paper we prove the existence of a weak solution to a doubly nonlinear
parabolic fractional $p$-Laplacian equation, which has general doubly
non-linearlity including not only the Sobolev
subcritical/critical/supercritical cases but also the slow/fast diffusion ones.
Our proof reveals the weak convergence method for the doubly nonlinear
fractional $p$-Laplace operator.
