Lambert $\boldsymbol{W}$ function method in investigating asymptotic properties of~fractional~delay differential equations

Typically, stability/asymptotic issues connected with the linear delay differential equation \begin{align}\label{nechvatal:eq1} x'(t)=\lambda x(t-\tau),\quad \lambda\in\mathbb{C}, \tau>0, \end{align} can be treated by the Lambert $W$ function method. Under some restrictions, this approach can still be used in the case of a fractional counterpart of the mentioned equation (i.e., when the first-order derivative on the left-hand side is replaced by a fractional derivative $D^\alpha x$ of a suitable order $\alpha$). The key step of our considerations consists in the fact that the Lambert $W$ function (despite it is a complex function) can be, in some sense, manipulated in the real domain only. Then, we are very easily able to rediscover the known ``iff'' condition on $\lambda$ for the asymptotic stability of the zero solution to the fractional version of \eqref{nechvatal:eq1}. In addition, a precise description of the decay/growth rate of the solutions can be obtained.