Resolvent Families for Multi-Term Caputo–Fabrizio Cauchy Problems with Damping

We construct a resolvent operator framework for multi-term fractional Cauchy problems with damping in Banach spaces, where the fractional derivatives are of the Caputo–Fabrizio (CF) type. Using Laplace transform techniques, we obtain necessary and sufficient conditions for the generation of the associated resolvent families and establish the existence and uniqueness of generalized solutions. Since the CF derivative vanishes at the origin, the classical initial condition u(0+)=x is unattainable when Ax≠0; the resolvent constructed here gives rise to a generalized solution with u(0+)=T(0)x≠x. We make this initial jump precise, show that it vanishes at the rate O(1−α) as α→1−, and indicate when it is physically meaningful. The abstract theory is applied to a biharmonic plate model with structural damping, producing explicit analytical solutions and a quantitative bound on the jump.