Suppose I have a system of delay differential equations
$\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$
$\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$
where $G_1, G_2, F_1, F_2$ are known well behaved functions, further simplification of the problem is untractable.
The domain of $\phi_1$ is $S=[a,b]\subset[0,1]$,
The domain of $\phi_2$ is $[1-b,1-a]$.
$\lim_{s\to a}G_1(\phi_1(s))$, $\lim_{s\to b}G_1(\phi_1(s))$,$\lim_{s\to b}G_2(\phi_2(1-s))$,$\lim_{s\to a}G_2(\phi_2(1-s))$ are bounded above and below
The problem is: given initial conditions
$\phi_1(a)=x$
$\phi_2(1-b)=y$
Is there any theorems to show that the solution to the system is unique?
Thanks.