Let $B=\begin{bmatrix} P & 0 \\ 0 & Q \end{bmatrix}$, where $P$ is $k\times k$ matrix whose eigenvalues have negative real part, and $Q$ is $(n-k)\times (n-k)$ matrix whose eigenvalues have positive real part.
Let $\dot y=By+G(y)$,
Why are the solutions $y$ to this equation, are also solutions to the following integral equation:
$$u(t,a)=U(t)a+\int_0^tU(t-s)G(u(s,a))ds-\int^{\infty}_t V(t-s)G(u(s,a))ds$$?
Where $U(t)=\begin{bmatrix}e^{Pt} & 0 \\ 0 & 0 \end{bmatrix}$, and $V(t)=\begin{bmatrix}0 & 0 \\ 0 & e^{Qt} \end{bmatrix}$