On the geometric nature of partial and conditional stability

We prove that certain problems which generalize the classical stability problem studied by A. M. Lyapunov admit a coordinate-free description. Namely, we mean problems on partial and conditional stability of solutions to vector functional differential equations, as well as a more general problem on the dependence of asymptotic properties of certain components of solutions on other ones.
For equations in the form
$$ x(t)-A\int^t_0x(s)d_sr(t,s)=f(t), $$
where the matrix $A=\mathrm{const}$ and $r:\{(t,s):0\le s\le t\}\to\mathbb C$, the indicated types of stability are defined by properties of minimal subspaces of the vector space which are invariant with respect to a given transformation and belong to a given subspace.