Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

We firstly prove that β‐times integrated α‐resolvent operator function ((α, β)‐ROF) satisfies a functional equation which extends that of β‐times integrated semigroup and α‐resolvent operator function. Secondly, for the inhomogeneous α‐Cauchy problem cDtαu(t)=Au(t)+f(t), t ∈ (0, T), u(0) = x0, u′(0) = x1, if A is the generator of an (α, β)‐ROF, we give the relation between the function v(t) = Sα,β(t)x0 + (g1*Sα,β)(t)x1 + (gα−1*Sα,β*f)(t) and mild solution and classical solution of it. Finally, for the problem cDtαv(t)=Av(t)+gβ+1(t)x, t > 0, v(k)(0) = 0, k = 0,1, …,N − 1, where A is a linear closed operator. We show that A generates an exponentially bounded (α, β)‐ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L loc 1(ℝ+,X). Our results extend and generalize some related results in the literature.