Let $x(t)$ satisfy the following Delayed Differential Equation (DDE): \begin{align*} x'(t)&=x(t)-1 &\mbox{ if } t \leq 1\\ x'(t) &= -x(t-1) (x(t-1) - x(t)) & \mbox{ if } t>1, \end{align*} let $x(0)=x_0 \in (0,1)$ be the boundary condition. One easily finds that on $[0,1]$ the solution is given by $x(t)=1-(1-x_0)e^t$. I would like to show the following claim:
If $x_0^{(1)} < x_0^{(2)}$ and $x^{(1)}, x^{(2)}$ are solutions with boundary condition $x^{(i)}(0)=x_0^{(i)}$ and for all $s<t$ we have $0<x^{(1)}(s), x^{(2)}(s)$, then for all $s<t$ we have $x^{(1)}(s) < x^{(2)}(s)$.