Non constant delay differential equations

Question

Given $\varphi:[0,1] \to [0,1]$ a continuous function, let $(E)$ be the delay differential equation (I am not sure about the terminology, as the delay is non constant): $y'(t) = y(\varphi(t))$. It is not hard to prove that the solutions form a finite dimensional space, by using Ascoli theorem; and in the case where $|\varphi(t)| \leq t$, the space of solutions is actually one dimensional (by Picard method). I wonder if anything more can be said, eg about the dimension of the space of solutions? Thank you!