Does globally stability implies existence of Lyapunov function? I recently had a chance to ask a professor this question and he mentioned that this is true for ordinary differential equations (but not for delay differential equation). He did not remember the reference, I could not find a reference for this, so can anyone please point me to the right source?

You will find what you need in this paper and the references of  Karafyllis:
Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form
Pierdomenico Pepe  & Iasson Karafyllis
