I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation

$$ \text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t) $$

with initial data $g(t)=g_0(t)$ on $-T \leq t\leq 0$, let's say $g_0\in\mathcal C^n$.

The asymptotics of solutions to

$$ \text{(2)} \quad\quad\quad g'(t)-Cg'(t-T)=Ag(t)+Bg(t-T), $$

including $C=1$, is governed by the characteristic function

$$ H(\lambda)=\lambda(1-Ce^{-\lambda T})-A-Be^{-\lambda T} $$

via the following theorem

If $\alpha_0=\sup\{Re(\lambda)\,:\,H(\lambda)=0\}$ and $g(t)$ is a solution to (2), then, for any $\alpha>\alpha_0$, there is a constant $K=K(\alpha)$ such that $$ > \lvert g(t)\rvert\le Ke^{\alpha t}\sup_{-T\le s\le0}\lvert g_0(s)\rvert. > $$

Because of the context in which equation (1) arises I expect $g(t)$ to converge to $0$, so let's apply the theorem.

The roots of characteristic function of (1) are solutions to equation

$$ e^{\lambda T}=\frac{\lambda}{\lambda+1}. $$

After taking the module of both sides we have

$$ e^{Re(\lambda) T}=\big\lvert\frac{\lambda}{\lambda+1}\big\rvert<1, $$

so $Re(\lambda)<0$. On the other hand, for $\lvert\lambda\rvert\gg1$ real part of $\lambda$ converges to $0-$, so $\alpha_0=0$ and the last theorem is inconclusive in the matter of convergence of $g(t)$ to $0$ so the question is:

*How to establish rate of convergence (or divergence) of solutions to
equation (1) and its (in)dependence on initial data?*

The conjecture is that $g(t)\sim t^{\gamma}e^{-\beta t}$, where $\gamma$ and $\beta\ge0$ might depend on the differentiability class of initial data, but that's a wild guess.

PS I already posted the question on math.stackexchange, only then I read about mathoverflow and found out that it would be more suitable, so sorry for posting it twice.