Is there a way of finding the condition required for a delayed differential equation to have a steady state solution? By 'steady state' I mean that $\frac{dx}{dt}=0$, or alternatively, $x(t)$ approaching some constant as $t \rightarrow\infty$.

If we have an ordinary differential equation:

$\frac{d}{dt}x(t) = ax(t) + c$

then we can solve it using the integrating factor method to get the following solution:

$x(t) = e^{at}x(0) + -\frac{c}{a}[1-e^{at}]$ 

assuming we have the initial condition $x(0)$. Call this equation (\*). We can now see that a steady state solution exists only if $a<0$, so all of the exponential terms become $0$ as $t \rightarrow \infty$. 

My question is this: is there an equivalent condition for a steady state solution to exist for a delayed differential equation of the form:

$\frac{d}{dt}x(t) = a x(t) + bx(t-\tau) + c$ ?

My hand-waving solution is that the condition is $(a +b)<0 $ for steady state to exist, since when $t \rightarrow\infty$, there isn't a difference between $t$ and $t-\tau$. This also recovers the correct answer in the case where $\tau \rightarrow 0$. I was hoping someone might be able to confirm or dis-confirm this in a slightly more rigourous way. Thanks!