The Fundamental Solutions for Fractional Evolution Equations of Parabolic Type

where 0 < α≤ 1, Γ(α) is the gamma function, {A(t) : t ∈ [0,T]} is a family of linear closed operators defined on dense set D(A) in a Banach space E into E, u is the unknown Evalued function, u0 ∈ D(A), and f is a given E-valued function defined on [0,T]. It is assumed that D(A) is independent of t. Let B(E) denote the Banach space of all linear bounded operators in E endowed with the topology defined by the operator norm. We need the following conditions. (A1) The operator [A(t) + λI]−1 exists in B(E) for any λ with Reλ≥ 0 and ∥∥∥[A(t) + λI]−1∥∥∥≤ C |λ|+ 1 , (1.2)