The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative

Abstract

Given an injective closed linear operator A defined in a Banach space (Formula presented.) and writing (Formula presented.) the Caputo–Fabrizio fractional derivative of order (Formula presented.) we show that the unique solution of the abstract Cauchy problem (Formula presented.) where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem (Formula presented.) where the family of bounded linear operators (Formula presented.) constitutes a Yosida approximation of A and (Formula presented.) as (Formula presented.) Moreover, if (Formula presented.) and the spectrum of A is contained outside the closed disk of center and radius equal to (Formula presented.) then the solution of (Formula presented.) converges to zero as (Formula presented.) in the norm of X, provided f and (Formula presented.) have exponential decay. Finally, assuming a Lipchitz-type condition on (Formula presented.) (and its time-derivative) that depends on (Formula presented.) we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set (Formula presented.).