Consider this system of differential equations for $t\in[0,\infty)$:
$$ \frac{d}{dt}x(t) = a(t) + F(x(t), y(t)),$$ $$ \frac{d}{dt}y(t) = a(t) + G(x(t), y(t)),$$
with positive initial conditions: $y(0)>0, x(0) >0$, where $a(t)$ is a piecewise continuous function and $F$ and $G$ are some continuous functions. Assume the system has a unique continuous solution $(x(t), y(t))$.
Fix $T>0$ and consider the interval $I=[0,T]$. Assume that on this interval there exists a sequence of (nice) continuous functions $\{a_n(t)\}_{n=1}^{\infty}$ converging to $a(t)$ in $L^1$ $\left(\int_{0}^{T} |a(t)-a_n(t)| dt \rightarrow 0\right)$, such that this system of differential equations:
$$ \frac{d}{dt}x_n(t) = a_n(t) + F(x_n(t), y_n(t)),$$ $$ \frac{d}{dt}y_n(t) = a_n(t) + G(x_n(t), y_n(t)),$$
with the same initial conditions $x_n(0)=x(0)>0$ and $y_n(0)=y(0)>0$, has positive continuous solutions $x_n(t)>0$ and $y_n(t)>0$ for $t \in I$.
$\bf{1 -}$ Is this information enough to prove that the solutions $x(t)$ and $y(t)$ of the original ODE system is nonnegative: $$ x(t) \ge 0, \quad y(t) \ge 0.$$ for $t \in I.$?