Resolvent Approach to Atangana--Baleanu Evolution Equations: Laplace Symbols, Mild Solutions, and Regularity

Fractional evolution equations with memory terms are widely used to model anomalous diffusion, viscoelastic response, and hereditary dynamics in physics, biology, and engineering. Among the recently introduced operators, the Atangana--Baleanu (AB) derivatives have attracted considerable attention due to their non-singular Mittag--Leffler kernels. However, their analytic treatment remains limited, as the AB kernel does not fall within the classical Volterra or Bernstein-function frameworks. This paper develops a unified resolvent approach for AB-type evolution equations in Banach spaces. Using a Laplace-domain formulation inspired by Hille--Phillips theory, we introduce a fractional resolvent associated with the AB kernel and establish optimal bounds on sectorial contours. Under the natural condition $\beta<1+\alpha$, we construct an AB--Mittag--Leffler resolvent family and obtain a complete representation of mild solutions to the AB Cauchy problem. Sharp stability and regularity estimates of Mittag--Leffler type are derived, including fractional-domain bounds. Numerical illustrations confirm the predicted decay, and connections with non-autonomous operators, maximal $L^p$-regularity, and weighted AB kernels are outlined. The results place AB-type equations within a functional-analytic framework comparable to the classical theory for Caputo and Volterra models.