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 ) } .