Consider a differential equation of the form $$ \dot \gamma(t)=F_t(\gamma(t)) $$ For $\gamma:[0,1]\to \mathbb R^{2n}$ and a smooth $F:[0,1]\times \mathbb R^{2n}\to \mathbb R^{2n}$ such that every $F_t$ is affine linear.
Then for every initial value $x_o\in\mathbb R^{2n}$ there exists a unique solution $\gamma$ with $\gamma(0)=x_0$. (right?)
But what about solutions with boundary values on the first $n$ components? Does there exists a solution with $$ \gamma(0)_i=0=\gamma(1)_i \text{ for } i=1,\dots,n? $$
Rephrased: Consider the map $$E:\mathbb R^{2n}\to \mathbb R^{2n},x_0\mapsto \gamma_{x_0}(1)$$ where $\gamma_{x_0}$ is the unique solution with $\gamma_{x_0}(0)=x_0$. Now what is the image $E(\{0\}^n\times \mathbb R^n)$? Does it intersect $\{0\}^n\times \mathbb R^n$?
Context: This question comes from reducing a second order linear DE to a first order linear DE. There I want a solution of this 2nd order DE given $\gamma(0)$ and $\gamma(1)$ as opposed to given the initial conditions $\gamma(0)$ and $\dot \gamma(0)$. (Sometimes called "Two-Endpoint problem")
I dont know anything about theory of differential equations; Any keywords or pointers to what to look for are appreciated
Edit: Just found this: When do boundary conditions specify a unique solution to ODEs?