Let X be a real Banach space. We consider the existence of solutions of the abstract nonlinear functional evolution equation : $$ (E) frac{du(t)}{dt} + A(t)u(t) + F(u)(t) i h(t), $$ $$ u(s) = x_o in D(A(s)), 0 leq s leq t leq T, $$ where u : $[s, T] o x$ is an unknown function, ${A(t) : 0 leq t leq T}$ is a given family of nonlinear (possibly multivalued) operators in X, and $F : C([s, t];X) o L^{infty}([s, X];X)$ and $h : [s, T] o X$ are given functions.