t_1, u(t_1; t_0, a))$ that starts at the point $u(t_1; t_0, a)$ at the moment $t_1$, we obtain a spliced solution of , \[splice\] w(t; t\_0, a) = u v(t) =

u(t; t\_0, a), & t\_0tt\_1,\
v(t; t\_1, u(t\_1;, t\_0, a)), & tt\_1.

If the solutions of equation are unique, we have the following analog of the semigroup property: \[n-sg\] u(t\_0; t\_0, a) = a u(t\_2; t\_1, u(t\_1; t\_0, a)) = u(t\_2; t\_0, a),aXt\_2t\_1t\_00. Also, we can define the *transition* map $U(t_1; t_0)$ that maps $X$ into $X$ by assigning to every $a\in X$ the value $u(t_1; t_0, a)$ of the solution $u(t; t_0, a)$. This transition map has the properties

\[NSG:main\] $$\begin{aligned}
& U(t; t) = \hbox{id}_X\,,\;\forall t\ge 0 \label{NSG:a}\\
& U(t_2; t_1)\circ U(t_1; t_0) = U(t_2; t_0)\,,\;\forall t_2\ge t_1\ge t_0\ge 0\,.\label{NSG:b}\end{aligned}$$

A family of maps $U(t_1; t_0)$ with the properties will be called a *process* (see, e.g., [@Dafermos; @Hale]). To the autonomous case correspond homogeneous processes characterized by time invariance: $U(t_1-\tau; t_0 - \tau) = U(t_1; t_0)$ for all (admissible) $\tau$ (and therefore $U(t) = U(t; 0)$ is the semigroup).

It may be advantageous to think of solutions of the autonomous equation as integral curves (trajectories) in $X$, 