Growth Equation of the General Fractional Calculus

Abstract: We consider the Cauchy problem

(m a t h b b D_{(k)} u) (t) = l a m b d a u (t)

,

u (0) = 1

, where

m a t h b b D_{(k)}

is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {�f 71} (2011), 583--600),

l a m b d a > 0

. The solution is a generalization of the function

t m a p s t o E_{a} l p h a (l a m b d a t^{a} l p h a)

where

0 < a l p h a < 1

,

E_{a} l p h a

is the Mittag-Leffler function. The asymptotics of this solution, as

t o i n f t y

, is studied.

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