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 ẋ(t)=F(t,xt), where xt(θ)=x(t+θ)xt(θ)=x(t+θ) for all θ∈[−r,0]θ∈[−r,0] and F:[0,A]×X2→X,XF:[0,A]×X2→X,X is a Banach space and X2X2 is the Banach space of continuous functions defined on [−r,0][−r,0] with values in XX.