On general properties of $n$-th order retarded functional differential equations

Consider the second order RFDE (retarded functional differential equation) $x''(t) = f(t,x_t)$, where $f$ is a continuous real-valued function defined on the Banach space $\R \times C\sp{1}([-r,0],\R)$. The weak assumption of continuity on $f$ (due to the strong topology of $C\sp{1}([-r,0],\R)$) makes not convenient to transform this equation into a first order RFDE of the type $z'(t) = g(t,z_t)$. In fact, in this case, the associated $\R\sp{2}$-valued function $g$ could be discontinuous (with the $C\sp{0}$-topology) and, in addition, not necessarily defined on the whole space $\R \times C([-r,0],\R\sp{2})$. Consequently, in spite of what happens for ODEs, the classical results regarding existence, uniqueness, and continuous dependence on data for first order RFDEs could not apply.\\ Motivated by this obstruction, we provide results regarding general properties, such as existence, uniqueness, continuous dependence on data and continuation of solutions of RFDEs of the type $x\sp{(n)}(t)=f(t,x_t)$, where $f$ is an $\R\sp{k}$-valued continuous function on the Banach space $\R \times C\sp{(n-1)}([-r,0],\R\sp{k})$. Actually, for the sake of generality, our investigation will be carried out in the case of infinite delay.