I am having a really hard time to try to understand this problem. what is non local initial conditions and how it work (or how we can use it) in practical applications like, in physics and engineering ? Here is an example of system with nonlocal initial conditions: \begin{equation} \left\{\begin{array}{l c l} \frac{dx(t)}{dt} + Ax(t) = f(t, x(t)), \ t\in I=(t_0, t_0 + a],\\ x(t_0) + g(t_{1},..., t_p, x(.)) = x_0,&\\ \end{array}\right. \end{equation} with solutions when we are using semigroup theory maybe this : $$x(t)=S(t)x(t_0) + \int_0^tS(t-s)(s, x(s))ds,$$
$S(t)$ is semigroup. For example, the use of this non-local conditions (here the function $g$) is said to enable additional time measurements $_, = 1, 2,\cdots, $ which is more accurate than measuring at $ = 0$ alone and provides more information about the system.