Growth Equation of the General Fractional Calculus

We consider the Cauchy problem $(mathbb D_{(k)} u)(t)=lambda u(t)$, $u(0)=1$, where $mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {bf 71} (2011), 583--600), $lambda >0$. The solution is a generalization of the function $tmapsto E_alpha (lambda t^alpha)$ where $0<alpha <1$, $E_alpha$ is the Mittag-Leffler function. The asymptotics of this solution, as $tto infty$, is studied.