Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups

We investigate the local well-posedness of semilinear fractional integro-differential equations in Banach spaces with the Hadamard fractional derivative. The equation is DtβHu(t)=Au(t)+φt,u(t),∫1tK(t,s)ρ(s,u(s))ds,u(1)=u0, where A generates a compact C0 semigroup. Using Schauder’s fixed point theorem, we prove local existence under linear growth conditions. Uniqueness is obtained via Banach’s contraction principle under Lipschitz assumptions. The main contribution is a detailed theorem for non-Lipschitz nonlinearities satisfying Carathéodory conditions and Osgood-type growth, where we prove the existence and additional regularity of mild solutions. An illustrative example with Lipschitz nonlinearities is provided.