LAUSR

Abstract This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β-times integrated α-resolvent operator function ((α, β)-ROF) and it is injective, then the inverse operator A−1 is the generator of a tempered (α, γ)-ROF for all γ > β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A−1 is in addition the generator of a tempered (δ, 0)-ROF for all δ < α. Moreover, the optimal decay rate of (α, β)-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.