The authors consider the Cauchy problem \[ D^\beta_t u(t)+ Au(t)= f(t,u(t))+ \int^t_0 K(t-s) g(s,u(s))\,ds,\quad t\in [0,T], \] with \(u(0)= H(u)\). Here \(D^\beta_t\) is a fractional time derivative of order \(\beta\in(0, 1)\), \(-A\) generates a compact analytic semigroup \(T(t)\) on a Banach space \(X\), \(u\in C([0,T]; X_\alpha)\) (\(X_\alpha\) the domain of \(A^\alpha\), \(0<\alpha< 1\)); \(f,g: [0,T]\times X_\alpha\to X\), \(K\in C[0,T]\) and \(H: C([0,T]; X_\alpha)\to X_\alpha\). In the example, \[ H(u)= \sum^N_{i=1} \int^\pi_0 K_0(x,y)\cos u(t_i; y)\,dy,\quad 0\leq x\leq\pi. \] The authors first give a technical definition of mild solutions of the Cauchy problem, and then -- under conditions too lengthy to be included here -- prove the existence of a mild solution. The proof is by Schauders fixed point theorem.