The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative

Abstract

Given an injective closed linear operator A defined in a Banach space X, and writing CFDtα the Caputo–Fabrizio fractional derivative of order α∈(0,1), we show that the unique solution of the abstract Cauchy problem (∗)CFDtαu(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)=Bαu(t)+Fα(t),t≥0;u(0)=−A−1f(0), where the family of bounded linear operators Bα constitutes a Yosida approximation of A and Fα(t)→f(t) as α→1. Moreover, if 11−α∈ρ(A) and the spectrum of A is contained outside the closed disk of center and radius equal to 12(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−1f(0,x)}.