Semilinear Hadamard Fractional Integro-Differential Equations with Nonlocal Conditions and State-Dependent Delay Under Compact Semigroup

Abstract

This paper studies local and global existence, uniqueness, and Lipschitz continuous dependence of mild solutions for a semilinear Hadamard fractional integro-differential equation in a Banach space X, with nonlocal initial conditions, a state-dependent Volterra kernel embedded in the nonlinearity, and impulsive effects at finitely many fixed times, where −A generates a compact C0-semigroup on X. The mild-solution formula is derived from first principles via τ=lnt and the Riemann–Liouville variation of parameters formula. Local existence follows from Schauder’s fixed-point theorem, global existence from a Hadamard–Gronwall inequality with an explicit blow-up alternative, uniqueness from a logarithmic power-weight contraction with an explicit condition on the Lipschitz constants, and continuous dependence with an explicit stability constant Cdep; two concrete examples verify all hypotheses. Beyond this compact-semigroup theory, we prove two further results: an existence theorem under a Kuratowski measure-of-noncompactness condition via a Hadamard-adapted Mönch fixed-point argument, removing the compactness assumption on {T(t)}t≥1 altogether, and an Ulam Hyers Rassias stability theorem whose constant reuses the same contraction weight λ* from the uniqueness theorem. Both extend recent noncompact-semigroup and Ulam-stability results in the literature to the present nonlocal, impulsive, state-dependent-kernel setting.