An existence result for a functional differential equation on a Banach space

We use the nonlinear alternative and topological transversality to prove an existence theorem for the solutions of the functional differential equation x ̇ ( t ) = F ( t , x t ) , where x t ( θ ) = x ( t + θ ) for all θ ∈ [ − r , 0 ] and F : [ 0 , A ] × X 2 → X , X is a Banach space and X 2 is the Banach space of continuous functions defined on [ − r , 0 ] with values in X .