Growth equation of the general fractional calculus

Abstract

We consider the Cauchy problem (D(k)u)(t) = λu(t), u(0) = 1, where D(k) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583-600), λ > 0. The solution is a generalization of the function t mapping Eα(λtα), where 0 < α < 1, Eα is the Mittag-Leffler function. The asymptotics of this solution, as t → ∞, are studied.