The existence of a mild solution of the abstract functional differential inclusion
$$P(\phi )\{\begin{array}{ll}{u}^{\prime }(t)+Au(t)-F(t,u_t)\ni 0,& t\in I=[0,T]\\ u(t)=\phi (t),& t\in J=[-r,0]\end{array}$$
is obtained by a Filippov technique. Here $A$ is an operator such that $A+\omega \mathcal{I}$ is m-accretive for some $\omega \in {\mathbb{R}}^{+}$, and the multimapping $F(t,.)$ is $h(t)$-Lipschitzian. The result is applied to a nonlinear functional control problem.