Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions

A local existence and uniqueness result for the functional differential equation in a Banach space X (FDE) $$begin{gathered} xprime (t) in f(t)x(t) + g(t)x_t ,0 leqslant t leqslant T, hfill x_0 = phi in C( - R,0;X) hfill end{gathered} $$ is obtained, for the case where the operatorsf(t) satisfy only a local dissipativity condition and the operatorsg(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.