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 Banaeh space X x'(t) is an element of f(t) x(t) + g(t)x(t), 0 <= t <= T, (FDE) x(0) = phi is an element of C(-R, 0; X) is obtained, for the case where the operators f(t) satisfy only a local dissipativity condition and the operators g(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.