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 $t\mapsto E_{\alpha }(\lambda t^{\alpha })$ where $0<\alpha <1$, $E_{\alpha }$ is the Mittag-Leffler function. The asymptotics of this solution, as $t\to \infty$, is studied.