Perturbation theory for fractional evolution equations in a Banach space

The paper proves uniform well-posedness for the fractional Cauchy problem CD_t^α u = (A+B(t))u with u(0)=x, u′(0)=y under B ∈ C^1 and A generating a fractional cosine family, constructs the perturbed families C_α(·;A+B), S_α(·;A+B) via a Dyson–Phillips series based on T_α(t;A)=I_t^{α−1}C_α(t;A), derives the Volterra equation u=u_0+T_α∗(Bu), and uses fractional calculus identities to conclude the Caputo equation; all steps mirror the candidate’s approach. Specifically: the main theorem (Theorem 3.1) establishes existence, uniqueness, C^2-regularity in D(A), and the series/Volterra representation for u and for the perturbed families , with the Volterra-to-Caputo differentiation performed via the identity CD_t^α(T_α∗(Bu)) = (D^1_t C_α)∗(Bu)+Bu and D^1_t C_α = AT_α . The Mittag–Leffler bounds on C_α(·;A+B) and S_α(·;A+B) match the model’s estimates exactly . The definitions and resolvent-type identities the model invokes are standard in the paper’s preliminaries, including T_α=I_t^{α−1}C_α, D^1_t C_α = AT_α, and AI_t^α T_α x = C_α x − x, which justify the model’s “resolvent identities” . The inhomogeneous case (Theorem 3.2) is handled with the same series/Volterra strategy as in the model’s step (3)–(4) . Minor stylistic differences remain (e.g., uniqueness via small-time iteration in the paper vs. the model’s Neumann-series estimate), but they are equivalent in substance. Overall, the candidate solution faithfully reconstructs the paper’s argument.