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)(alpha) the Caputo-Fabrizio fractional derivative of order alpha is an element of (0,1), we show that the unique solution of the abstract Cauchy problem (*) (CF)D(t)(alpha)u(t) = Au(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(alpha)u(t) + F-alpha(t), t >= 0; u(0) = - A(-1) f(0), where the family of bounded linear operators B-alpha constitutes a Yosida approximation of A and F-alpha(t) -> f t) as alpha -> 1. Moreover, if 1/1-alpha is an element of rho(A) and the spectrum of A is contained outside the closed disk of center and radius equal to 1/2(1-alpha) then the solution of (*) converges to zero as t -> infinity, 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 alpha, we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S := {x is an element of D(A) : x = A(-1 )f(0, x)}.