We consider a class of doubly nonlinear history-dependent problems associated with the equation



\partial_t k \times  (b(v) − b(v_0)) = \mbox{div }a(x, Dv) + f.



Our assumptions on the kernel $k$ include the case $k(t) = t^{−\alpha}/Γ(1−\alpha)$, in which case the left-hand side becomes the fractional derivative of order $\alpha \in (0, 1)$ in the sense of Riemann-Liouville. Existence of entropy solutions is established for general L$^1$−data and Dirichlet boundary conditions. Uniqueness of entropy solutions has been shown in a previous work.