Equivalence of different hereditary structures in ordinary differential equations (Q1268082)

The authors consider three hereditary structures: \[ \begin{aligned} x'(t) & = f(t,x_t),\quad t\in (t_0,t_0+ p],\\ x(t) & = \varphi(t),\quad t\in (-\infty, t_0],\end{aligned}\tag{P\(_1\)} \] where \(x_t: (-\infty, 0]\to \mathbb R^n\) is defined by \(x_t(\tau)= x(t+\tau)\): \[ \begin{aligned} x'(t) & = g(t, R(t,\Gamma(x, \alpha(t)))),\quad t\in [t_0,t_0+ p],\\ x(t) & = \varphi(t),\quad t\in I(t_0, p),\end{aligned}\tag{P\(_2\)} \] where \(\alpha\) is a multifunction with \(\alpha(t)\in (-\infty, t]\), \(\Gamma(x, \alpha(t))\) is the graph of \(x\) restricted to the set \(\alpha(t)\), \(R\) is a Volterra-type operator; \[ \begin{aligned} x'(t) & = h(t, x),\quad t\in [t_0, t_0+ p],\\ x(t) & = \varphi(t),\quad t\in (-\infty, t_0],\end{aligned}\tag{P\(_3\)} \] where \(h(t,\cdot)\) is a Volterra-type operator. The authors prove the main theorem: Every problem \(P_j\) admits an equivalent formulation as a \(P_j\)-type problem, \(i,j= 1,2,3\), \(i\neq j\), in such a way that there exists a monotone homeomorphism between the corresponding sets of solutions.