The aim of this paper is to present some remarks concerning the functional differential equation$$ v'(t)=G(v)(t) $$in a~Banach space $mathbb{X}$, where $G:cabxoabx$ is a continuous operator and $cabx$, resp. $abx$, denotes the Banach space of continuous, resp. Bochner integrable, abstract functions.It is proved, in particular, that both initial value problems (Darboux and Cauchy) for the hyperbolic functional differential equation$$ rac{partial^2 u(t,x)}{partial t,partial x}=F(u)(t,x) $$with a Carathéodory right-hand side on the rectangle $[a,b]imes[c,d]$ can be rewritten as initial value problems for abstract functional differential equation with a suitable operator $G$ and $mathbb{X}=ccdr$.
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