Fractional Evolution Equations Governed by Coercive Differential Operators

and Applied Analysis 3 2. α-Times Regularized Resolvent Family Throughout this paper, X is a complex Banach space, and we denote by B X the algebra of all bounded linear operators on X. Let A be a closed densely defined operator on X, let D A and R A be its domain and range, respectively, and let α ∈ 0, 2 , C ∈ B X be injective. Define ρC A : {λ ∈ C : λ − A is injective and R C ⊂ R λ − A }. Let Σθ : {λ ∈ C : |arg λ| < θ} be the open sector of angle 2θ in the complex plane, where arg is the branch of the argument between −π and π . Definition 2.1. A strongly continuous family {Sα t }t≥0 ⊂ B X is called an α -times C regularized resolvent family for A if a Sα 0 C; b Sα t A ⊂ ASα t for t ≥ 0; c C−1AC A; d for x ∈ D A , Sα t x Cx ∫ t 0 t − s α−1/Γ α Sα s Axds. {Sα t }t≥0 is called analytic if it can be extended analytically to some sector Σθ. If ‖Sα t ‖ ≤ Me t ≥ 0 for some constants M ≥ 1 and ω ∈ R , we will write A ∈ CαC M,ω , and C α C ω : ∪{C α C M,ω ;M ≥ 1}, C α C : ∪{C α C ω ;ω ≥ 0}. Define the operator à by Ãx C−1 ( lim t↓0 Γ α 1 tα Sα t x − Cx ) , x ∈ D ( à ) , 2.1