The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative.

Given an injective closed linear operator A defined in a Banach space X , and writing C F D t α the Caputo–Fabrizio fractional derivative of order α ∈ (0 , 1) , we show that the unique solution of the abstract Cauchy problem (∗) C F D t α u (t) = A u (t) + f (t) , t ≥ 0 , where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem u ′ (t) = B α u (t) + F α (t) , t ≥ 0 ; u (0) = − A − 1 f (0) , where the family of bounded linear operators B α constitutes a Yosida approximation of A and F α (t) → f (t) as α → 1. Moreover, if 1 1 − α ∈ ρ (A) and the spectrum of A is contained outside the closed disk of center and radius equal to 1 2 (1 − α) then the solution of (∗) converges to zero as t → ∞ , in the norm of X, provided f and f ′ have exponential decay. Finally, assuming a Lipchitz-type condition on f = f (t , x) (and its time-derivative) that depends on α , we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S : = { x ∈ D (A) : x = A − 1 f (0 , x) }. [ABSTRACT FROM AUTHOR]